Results of Culler and Shalen together with tameness, density, etc. imply that there exists a number $V$ such that if $M$ is a hyperbolic 3-manifold of volume $>V$, then the Margulis constant of $M$ is $\geq \log(3)$. This indicates that one ought to be able to compute the Margulis constant, making it an a priori trivial problem. To prove this, you take a sequence of 2-generator groups realizing the Margulis constant for manifolds with volume approaching $\infty$. In the limit, the Margulis constant is $>\log(3)$ by Culler-Shalen, so one concludes that there is some bound on volume for manifolds with Margulis constant $<\log(3)$.
Addendum: Recently Yarmola (in joint work with Futer and Gabai) has announced that they can show that the Margulis constant $\mu_3 > .5$.