(This is essentially a very long comment...)
What is r?
If your original $V$ is a curve, or alternatively after (as per Filipe) cutting your space down to a curve by taking hyperplane sections, then one has recourse to the following basic fact of algebraic geometry
normalization of curves: given any (possibly singular) curve algebraic curve $C \subset \mathbb{C}^n$, there is a unique map $\nu: C^\nu \to C$ such that $C^\nu$ is smooth, $\nu$ is surjective, and is a bijection away from the singularities of $C$.
So locally at some $p^\nu \in C^\nu$ mapping to your point of interest $p \in C$ you can choose a coordinate $z$ and write $$\nu(z) = (\nu_1(z), \cdots, \nu_n(z)) = (z^{a_1} + \ldots, z^{a_2} + \ldots, z^{a_3} + \ldots)$$
Filipe was explaining above that if you wanted something with a leading linear term, you have no choice but to write $z^{\mathrm{min}(a_i)} = w$. Note that this minimum doesn't depend on the coordinate $z$ you chose.
But what is this minimum?
If $V$ is a unibranch* curve, then $r = \mathrm{mult}_p(V)$, the multiplicity of $V$ at $p$.
The multiplicity of a variety at a point has various characterizations. If $V$ were cut out by one equation and $p=0$, then it's just the degree of the minimal degree monomial in that equation. In general, it's
- The number of intersections, near $p$, of $V$ with a general hyperplane passing near $p$
- The same for the tangent cone of $V$ at $p$
- The number of copies of the exceptional divisor when you blowup $V$ at $p$
- Take $\mathcal{O}_p, \mathfrak{m}_p$ to be the local (or complete local) ring of functions of $V$ at $p$, and its maximal ideal. Then the quantity $HS(n) = \mathrm{dim}_\mathbb{C} \mathcal{O}_p / \mathfrak{m}_p^{n+1}$ is called the Hilbert-Samuel function; it's eventually a polynomial in $n$ with leading term $n^{\mathrm{dim}(V)}/\mathrm{dim}(V)!$ times the multiplicity.
We can use that last characterization to see $\mathrm{mult}_p(C) = \mathrm{min}(a_i)$: the point is that the completed ring of functions $\mathcal{O}_p$ is the subring of $\mathbb{C}[[z]]$ generated by the $\nu_i(z)$. One sees from the fact that the normalization is generically 1-1 that $\mathrm{gcd}(a_i) = 1$, from which it follows that sufficiently high powers $\mathfrak{m}_p^N$ will just be $z^{N \cdot \mathrm{min}(a_i)} \mathbb{C}[[z]]$.
What is this word unibranch?
Well, when you normalize a curve, the preimage of the point $p$ might be several points $p^\nu_i$, the neighborhood of each of which maps to a different analytically locally irreducible component of the (reducible) analytic local curve $C$. One would like to say: just treat each local irreducible component separately. However for the exact question you asked, this isn't OK -- one could for example have a connected rational curve with a unique singularity $p$ such that $C$ at $p$ is analytically locally of the form $x(y^2- x^3) = 0$. This has two branches, one smooth ($x = 0$), and the other singular ($y^2 - x^3 = 0$). For any point on the curve, including points arbitrarily close to $p$ along the singular branch, one can get to them by traveling only along the smooth branch. Thus for this curve, the minimal $r$ is $1$, despite the presence of the singularity. I suspect this means you've asked slightly the wrong question, and should demand that your paths `never travel too far away from $p$', but I'm not exactly sure what's the best way to make that precise.
(You demanded your paths stay in a small open set $U$, which would fix the problem if $V$ is a curve. But, it could be that $V$ is something like the cone over the curve I described above, and you are looking at the cone point, in which case you'll see something like this behavior in any neighborhood.)
What if I just want to bound $r$?
Then Filipe's answer tells you the following: $r$ is less than the maximal multiplicity of any curve obtained by cutting the variety by an appropriate-dimensional linear slice. (I see no a-priori reason to believe that linear slices give better bounds than some other kind of slices, but also as mentioned above I don't know how to correctly formalize the question so that `what is $r$' actually makes sense.)
I do not know the name of this number -- i.e., the maximal multiplicity of a linear-slice-down-to-a-curve -- in algebraic geometry.
A related thing which does have a name is the answer to this question for a general $q$ near $p$, i.e. the multiplicity of a general linear-slice-down-to-a-curve. Such a curve is called a local polar curve and its multiplicity is called a local polar multiplicity; people study these things.
I do not think either of these things can be computed from the tangent cone.
