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.
From a practical viewpoint, computability and the availability of software may restrict your choice of prior in any case.
See: