One of the tough things about this decision is that BMC and The Prom are two incredibly divisive musicals: you either get them or you don't. As for "family-friendly," I'd probably lean towards The Prom, but depending on the social awareness of your 10-year-old, BMC might also be a good option. There are kids I know who are 9 that are listening to it.
"I think that when a movie says it was 'based on a true story,' oh, it happened - just with uglier people." - Peanut Walker, Shucked