Is there a q-analog to the braid group? The braid group $B_n$ on $n$ strands fits into a short exact sequence of groups:
$$ 1 \longrightarrow P_n \longrightarrow B_n \longrightarrow S_n \longrightarrow 1,$$
where $S_n$ is the symmetric group on the strands, and $P_n$ is the normal subgroup of braids that do not permute the strands.
Since symmetric groups are, in some sense, ``general linear groups over the field with one element,'' perhaps there is some corresponding short exact sequence ending with $GL_n(F_q)$ that specializes to the exact sequence above as $q \rightarrow 1$.
In the spirit of the exact sequence above, is there a $q$-analog to the braid group?
 A: You may want to look at these slides of Jon McCammond's talk. I am not sure he wrote a paper about it, but he did introduce the idea similar to what you want and in more general situation (arbitrary Artin group instead of the braid group), 
A: I don't see why there would be a unique scheme that "converges" to the braid group as $q\to1$. Here are some groups that could reasonably considered as the $q$-analogues of $B_n$:


*

*Consider the non-commutative power series algebra over $\mathbb F_q$ in $n$ variables, usually written $A=\mathbb F_q\langle\langle x_1,\dots,x_n\rangle\rangle$. Its group of automorphisms $B_{n,q}$ is a $q$-analogue of $B_n$; the natural map $A\mapsto A/\langle x_ix_j\rangle$ induces the map $B_{n,q}\to GL(n,q)$.

*At least when $q$ is prime, consider the subgroup of $A$ generated by $1+x_1,\dots,1+x_n$. A classical result of Magnus says it's a free group on $n$ generators. Take its closure in $A$ --- that's a free pro-$q$ group. Consider then the subgroup of $B_{n,q}$ that preserves that group. (If $q$ is a prime power, presumably throw in a few more automorphisms to obtain a group mapping onto $GL(n,q)$).
There's a lot of theory on the following filtration of $B_n$: it has a subgroup $P_n$, as the poster mentioned; and $P_n$ has a lower central series converging to $1$, for which the structure of the successive quotients are well understood, as $S_n$-modules. A good $q$-analogue should probably have a filtration by $GL(n,q)$-modules with the same shapes and multiplicities in each degree.
A: I suggest you look at the Iwahori-Hecke algebras of type A.  These deform the symmetric group algebras with relations that look like
$T_i^2 = q + (1-q)T_i$
for generating elements $T_i$.  The braid relations
$T_iT_{i+1}T_i = T_{i+1}T_iT_{i+1}$
still hold though so you get a (surjective) morphism
$kB_n\rightarrow \mathcal{H}_n\rightarrow 0$
(From here on I'm less sure of the details)
giving you a `short exact sequence'
$0\rightarrow K_n\rightarrow kB_n \rightarrow \mathcal{H}_n\rightarrow 0$.
To define the kernel you should look at the coinvariants of $kB_n$ w.r.t. the coalgebra map of $\mathcal{H}_n$.  This makes $K_n$ an algebra but not necessarily a Hopf algebra (although it may be a braided Hopf algebra in a suitable category).
The Hecke algebra may be the algebra that you want because the q parameter counts the way the Borel double cosets in some $GL_n(k)$ multiply (recall the Bruhat decomposition).  When the Borels become trivial then $q$ becomes one.
But notice also that this deformation does not require a deformation of the braid group.  I have no idea what the algebra $K_n$ looks like and if indeed it is well defined, it may still turn out to be $kP_n$.
To offer an answer to your final question: there may very well be q-analogues of the braid groups, but they may not be what you should be looking for.
A: There is a nice paper by Etingof and Rains about a non-standard q-deformations of S_n:
http://xxx.lanl.gov/abs/math/0409261
They deform not quadratic relations but the braid relations.
It seems to be relevant to the discussion.
A: Possibly. 2 ideas in this circle are:
(1) The Artin braid groups can be formulated as ``Weyl groups" (corresponding to the $A_n$ Dynkin diagrams), and Weyl groups have q-analogues. Many references exist, but I'm not sure what the best is (hopefully others will know).
(2) A second but related perspective can be found in Braids, Q-binomials and Quantum Groups by M Aguilar.
A: Here is one strategy that has not been suggested: the braid group is the fundamental group of the space $U/W$, where $U$ is the set of points in the (complexified) reflection representation of $W=S_n$ that have trivial fixer.  Now $W=GL_n(\mathbb{F}_q)$ is a reflection group over the finite field $\mathbb{F}_q$; let $U$ be the set of points (over an algebraic closure $F$ of $\mathbb{F}_q$) in the reflection representation of $W$ that have trivial fixer and define the "braid group" to be the etale fundamental group of $U/W$.  Essentially by definition it has a surjection onto $W$.  It's not clear to me in what sense this q-braid group might converge to the usual one as q goes to 1, but it should definitely play an important role in the study of $GL_n(\mathbb{F}_q)$.  For instance, one might be able to define a Hecke algebra deforming $F[GL_n(\mathbb{F}_q)]$ via monodromy representations of this braid group. 
