"... he's gone to one of his student's birthday party."

I would be inclined to use students'. When you make the phrase one of his students possessive, students is still plural, so you stick the apostrophe after. Rather than above, where it looks like you started with one of his student and made that possessive. And we don't really say things like "Jane was one of his student." But the rewriting for clarity option is probably the wisest.