How are constants/functions named after their discoverer? In general, when a paper references an object discovered/defined in another paper by author X, it goes something along the lines of:
"Let $\tau$ be the constant defined by X in 1999 [1]$\ldots$",
or
"Let $f_{\mu}$ denote the function that generalizes the case $\ldots$ (X, [1])".
At what point does the literature start talking about "X's constant" or "The X function"? 
Who/what determines that an object discovered by somebody deserves to take his name?
 A: This is determined by informal consensus of researchers in an area. Anyone can propose a name for a mathematical object, just by using this name in a paper. Then the proposed name either sticks to the object or not. This depends on the opinions of other people working in the area. Eventually the name of the object becomes established.
Sometimes several names for the same object become established, and all of them popular.
For example Fatou set is the same as the set of normality. The name Fatou set
was proposed in two influential surveys in 1980s, and many people in the area use it.
The name does not have to be the name of the discoverer of the object.
For example, Drinfeld introduced an object which he named Yangian,
in honor of Chen-Ning Yang.
M. Lyubich and I gave the name "Baker's domain" to an object that we defined. 
Unlike some other "personal names" we proposed in the same paper, this one is used by everyone who writes on the subject. Arnold once stated a principle that
"If the thing is named after someone, this indicates that the person had nothing to do with the thing". M. Berry remarked that "Arnold Principle applies to itself".
Sometimes a multitude of names reflects a priority dispute or nationalistic feelings (Young diagram vs Ferrers diagram, Schwarz inequality vs Cauchy or
Bouniakowski etc.) The name Casorati–Sokhotski–Weierstrass theorem reflects
some priority research, but the theorem in question was earlier stated by Briot and Bouquet. Same happened with "Gauss-Manin connection" which was in fact discovered by Legendre.
Sometimes the accepted name is changed ($\pi$ used to be "Archimedes number" and
$e$ used to be "Euler's number").
A funny story happened with "Koebe constant". It turned out that it is equal to $1/4$, so one mathematician wrote "Now it cannot be called Koebe constant anymore,
because it turns out that it already has a name, namely $1/4$").
Sometimes a name is based on a mistake. For example Abel's equation
is the accepted name for $f(x+1)=g(f(x))$. It is based on a manuscript of Abel, but the manuscript itself turns out to be just Abel's personal notes on
the work of Napier, where the equation was introduced.
People usually do not give their own names to mathematical objects. However,
in his influential book on functional analysis, Banach introduces 
"spaces of type H", "spaces of type F" and "spaces of type "B". As his "spaces of type H and F" are nothing but Hilbert and Frechet spaces, people probably understood the hint:-)
G. Julia intensively lobbied (not publicly, in private correspondence, which is now published) that the "irregular set" be called Julia set. He succeeded.
Remark. I just found the paper, arXix:1204.4716v1 (Bernard Ycart, A case of mathematical eponimy: the Vandermonde determinant), which shows that there is a whole research area related to this question: it is called "mathematical eponimy". The paper cites several other publications on this subject.
