When did the abuse of notation $y=y(x)$ start? It's quite common nowadays to name a function and the application of the function to its input with the same letter. (Possibly more so in applied areas. Certainly many calculus textbooks do this.) 
When did this practice start?
In particular, did any of the old masters like Newton, Leibniz, Euler etc. ever write something like $y=y(x)$? 
Clarification: The question is really about the history of this practice. With whom did it start? 
I didn't want to discuss merits or demerits of this notation. If you want to provide a non standard interpretation of $y=y(x)$, please also back it up with historical references.
 A: I have seen this type of notation as a help in understanding quantifiers.  
Example.  Here is a statement (Bertrand's Postulate):

for every $k > 1$ there is a prime $p$ such that $k \le p < 2k$.

This may be written, to emphasize that $p$ depends on $k$, as:

for every $k > 1$ there is a prime $p = p(k)$ such that $k \le p < 2k$.

A reader can tell that $p$ depends on $k$ in the first one as it is. But putting $p = p(k)$ in the second one emphasizes that fact.
A: Warning. This is an attempt at an answer out of curiosity rather than an expert answer. 
Newton has the following passage in "Recomputation of surfaces of
least resistance," (1694) (see Whiteside*, pp. 470-471): 

Unde $aabb - 2aabx+aaxx+ bbxx = aay + xxy$
[capiendo fluxiones]
$- 2aab\dot x + 2aax\dot x+2bbx\dot x =2x\dot xy + aa\dot y+xx\dot y$

Whiteside (ibid) writes: "The dotted letters in immediate sequel are Newtonian fluxions; that is, $\dot x = \frac {dx}{dt}$ and $\dot y = \frac {dy}{dt}$ where t is some independent variable of ‘time’."
I'd like to add that I don't think that interpreted in the context (whether historical or modern), something like $x=x(t)$ (say in parametric equations) or $y=y(x)$ (say when** $y$ represents the distance from the $x$-axis at a certain $x$), would be an "abuse of the notation". 
*The mathematical papers of Isaac Newton Volume VI 1684-1691
** I am pretty sure I have seen something like this in historical texts, but I couldn't remember where. 
A: I don't feel that $\ f=f(x)\ $ is an abuse of notation. It is rather a message. When we have an expression like $\ f\ :=\ t^2\!\cdot\! x + s,\ $ then $\ f=f(x)\ $ means that in the future when we write $\ f'\ $ then it means $t^2$ and not $\ 2\cdot t\cdot x\ $ nor simply $\ 1.$ Otherwise, the announcement $\ f=f(x)\ $ doesn't really enter the rest of the proceeding. (Am I right?)
This notation and the calculations which follow it feels to us old-fashioned in the so-called pure mathematics because this kind of mathematical analysis appears much less these days than in the past.
We can still talk about an abuse, why not, but then there are many other convenient abuses.
A: Regarding the original question of who started literally writing $y=y(x)$ or something like it, which I understand Jacobi didn't do in the quoted 1840 paper: Cayley (1859, p. 3),

where $\Omega$ is regarded as a function of $r,v,y,$ or (as this may be expressed) where $\Omega = \Omega(r, v, y)$

sounds like an early example, in that he feels the need to explain the notation.
A: This seems to have started with Jacobi around 1840, when he re-introduced the popular notation for partial derivatives in De determinantibus functionalibus. He even provides a well-intentioned justification for starting this abuse of notation: he wanted a less ambiguous notation.
While reading Jacobi one should bear in mind, that during his time the $f$ in $f(x)$ was not yet officially called the function. This modern use only started after 1900 and after Frege, Dedekind, Peano and Cantor. (For more on this see Who first considered the $f$ in $f(x)$ as an object in itself, and who decided to call it a function?.) Prior to 1900 what was called the function was the $f(x)$ or the $y$ in $y=f(x)$, in other words the variable quantity, and not the "rule" $f$. This is also how Jacobi uses the word function below. Beware hence, that what he denotes with $f$ and calls a function, is actually a variable quantity. (In modern terminology: his $f$ is an object of type $\mathbb{R}$ and not of type $\mathbb{R}\to\mathbb{R}$.)
What follows is my translation of the German translation Ueber die Functionaldeterminante, 1941 p.2 ff:

Before I come to the actual subject matter, I will start with some
  remarks concerning the notation of partial differentials. And since
  this treatise will contain repeated talk about functions, which may or
  may not depend from one another, it seems appropriate to also add some
  elementary considerations about these. 
$$2. $$  
To distinguish the partial differentials from the  ordinary ones,
  hence from those where all variable quantities are seen as functions
  of a single variabel, Euler and others put the partial differentials
  in between brackets. But since an accumulation of brackets becomes
  rather annoying for reading and writing, I have preferred to use the
  characteristic $$ d  $$ to denote ordinary differentials and the
  characteristic $$ \partial $$ for partial differentials. Adopting this
  convention rules out misunderstandings. So if $f$ is a function of 
  $x$ and $y$, I will write
  $$ df =\frac{\partial f}{\partial x}dx +\frac{\partial f}{\partial y}dy. $$
Whenever a function contains only a single variable, one may use the
  characteristic $d$ or $\partial$ indifferently. 
  [...]
In order for the partial differentials, of a function which depends on
  more than one variable, to be completely determined, it does not
  suffice to provide the function to be differentiated and the variable
  with respect to which to differentiate; one must moreover express
  which quantities remain constant during the differentiation. For
  suppose $f$ is a function of $x,x_{1},\dotsc, x_{n}$. Take $n$
  arbitrary functions $\omega_{1}, \omega_{2}, \dotsc, \omega_{n}$   of
  these variables and consider $f$ as a  function of the variables
  $x,\omega_{1}, \omega_{2}, \dotsc, \omega_{n}$. Then, when
  $x_{1}, \dotsc, x_{n}$ remain constant, the
  $\omega_{1}, \omega_{2}, \dotsc, \omega_{n}$  will no longer be
  constant with changing $x$, and neither will, when
  $\omega_{1}, \omega_{2}, \dotsc, \omega_{n}$  remain constant, the
  $x_{1}, \dotsc, x_{n}$ stay constant. The expression $\frac{\partial f}{\partial x}$ will hence assume completely different values,
  depending on whether these or those quantities are assumed constant
  during differentiation. 

(In modern terminology one could rephrase this as follows: the vector field $\frac{\partial}{\partial x}$ associated to a coordinate chart $x,x_1,\ldots,x_n$ on a manifold, does not depend on the coordinate function $x$ alone, but also on the remaining coordinate functions $x_1, \dotsc, x_n$.)

Suppose for example we introduce for a function $f$ of the two
  variables $x$ and $y$, an arbitrary function $u$ of $x$ and $y$,  as
  the second independent variable instead of $y$.  Then the differential
  that was previously denoted by
  $$ \frac{\partial f}{\partial x} $$
  will now be expressed as 
  $$ \frac{\partial f}{\partial x} +
 \frac{\partial f}{\partial u}\cdot\frac{ \partial{u}}{{\partial x}},
 $$ so that the same signs $\frac{\partial f}{\partial x}$ denote
  completely different values, depending on whether $y$ or $u$ are kept
  constant, while differentiating  $f$ with respect to $x$. 
I will therefore in this treatise,  whenever partial differentials are
  needed, not only express with the statement: $f$ is a function of
  $x, x_{1}, \dotsc, x_{n}$, that $f$ depends on these variables,  hence
  that it remains constant when these are constant, and changes, when
  they change — that would equally be valid, if instead of
  $x, x_{1}, \dotsc, x_{n}$ any other variables $\omega, \omega_{1}, \dotsc,
 \omega_{n}$ functions of these, were introduced as independent
  variables — rather when I say, $f$ is a function of the 
  $x, x_{1} , \dotsc, x_{n}$ I want the following to be understood: 
  whenever this function is partially differentiated, the
  differentiation should occur in such a way, that of these variables
  only one changes while the others remain constant. 
Further, if the formulas are to be free of ambiguities, the notation
  should not only indicate the variable with respect to which the
  differentiation is occurring,  but also the whole system of
  independent variables, the function of which is being differentiated,
  so that one may recognise which quantities remain constant during
  differentiation.  And this is all the more necessary, since it is
  unavoidable that in the same calculation or even in one and the same
  formula, there appear partial differentials that refer to different
  systems of independent variables, for example in the above expression
  $$ \frac{\partial f}{\partial x} + \frac{\partial f}{\partial u}\cdot
 \frac{\partial{u}}{{\partial x}} $$ in which $f$ is seen as a function
  of $x$ and $u$, while $u$ is seen as a function of $x$ and $y$; this
  expression was precisely the one $\frac{\partial f}{\partial x}$
  transitions into, when $u$ is introduced as independent variable
  instead of $y$. But if we write next to the dependent variable all the
  independent variables to which the partial differentiation refers,
  then this expression can be depicted by the following formula, which
  is free of any  ambiguity:  $$ \frac{\partial f(x,y)}{\partial x} = \frac{\partial f(x,u)}{\partial x} +
 \frac{\partial f(x,u)}{\partial u}\cdot
 \frac{\partial{u(x,y)}}{{\partial x}}. $$

(To stress: he is using the same notation as function application, but he doesn't want to denote function application with it, he only wants to make explicit which variables are to be kept constant.)

Certainly this notation, as well as every other imaginable notation
  that would allow one to completely determine any partial
  differentiation from the symbols alone, would become very cumbersome
  in more general investigations or more involved formulas, yes even
  impracticable, since with higher numbers of independent variables and
  more terms it might happen that a formula, which can be expressed in a
  single line, takes up a whole page. Certainly one should place the
  highest value on a notation that eliminates any ambiguity, and which
  makes any formula understandable on its own, without oral
  clarifications. But when it was possible without too much disadvantage,
  and in view of the enormous and unavoidable verbosity of the notation,
  I settled for the shorter notation of differentials, that dispenses
  with the specification of the independent variables.

Up to this point one might argue that his notation is justified. But he starts doing it in places where I see no reason for it. For example further down he writes

Let $f, f_1, f_2$ etc. be mutually independent functions of the variables $x, x_1, \ldots, x_n$, and let $x$ be a variable contained in $f$ [...] it follows that the $m+1$ equations
  \begin{align}
f(x,x_1,\ldots,x_n)&=\omega\\
f_1(x,x_1,\ldots,x_n)&=\omega_1\\
&\ldots\\
f_m(x,x_1,\ldots,x_n)&=\omega_m
\end{align}
  do not determine more than $m+1$ quantities.

Here these $f$ are again variables quantities of type $\mathbb{R}$ and not of type $\mathbb{R}\to \mathbb{R}$, and there is no need to write $(x, x_1, \dotsc, x_n)$ next to the $f$'s in the equations.
I haven't been able to find similar "typing errors" in Bernoulli, Euler, Lagrange, Laplace, Gauss or Cauchy. Even after Jacobi there are several people like Riemann or Peano where I can't find this. Although it is easier to find in the second half of the 19th century. For example in Hermite or Maxwell and later in Felix Kleins lectures on mechanics as well as in Sommerfeld.
But since I have only looked at 1 or 2 works of each of the above authors, this answer is by no means conclusive.
