[Kupershmidt](https://arxiv.org/abs/math/0001187) argues for 
$$\mu_{K}([j]_q)\propto(pq)^j,\;\;[j]_q=\frac{q^j-1}{q-1},$$ 
as the "natural" $q$-geometric distribution, because it produces the $q$-analog of the Pascal (negative binomial) distribution. It has a simple normalization,
$$\mu_{K}([j]_q)=\frac{1-p}{1-(p;q)_\infty}\,(pq)^{j-1}.$$