The key point is to show that your analysis does not depend much on the prior in the first place.
You could try
- a conjugate prior (if one exists)
- a (possibly improper) uniform prior and
- a Jeffreys prior (justified by its invariance under re-parameterization)
and if you get answers that are reasonably close then that should provide some support that your analysis does not depend on arbitrary choice of prior.
BUGS and similar systems give quite a bit of flexibility regarding the prior but otherwise, from a practical viewpoint, easy computability and the availability of software may restrict your choice of priorin any case.