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Hamsters do eat grains but rice doesn't offer the best nutritional value & isn't really an appropriate ingredient in a mix when other better grains or pseudo grains can be used, you won't hurt your ham giving it a little rice now & then but I don't see that it would be good for your ham in the long run to use it on a regular basis in a mix.
What grains or pseudo grains are you using in your mix?
I think some people in Germany use wild rice when making their own mixes (if I remember rightly from previous research!) but that is not the same as the kind of rice humans usually eat, it's more like a grass seed therefore more appropriate.