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.