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.