I wore jeans to dining room almost EVERY night on both carnival and royal carribean. (except formal nights) I've never had anyone say anything to me. No one makes a big deal about it. Some girls have like 6 different skirts for every night at dinner..but I just wear jeans and usually a tee or hoodie.
You can wear whatever you want the rest of the day pretty much. Basically the only place with a dress code is the dining room..but I don't abide by it
and I'm not even a rebel or anything; I'm pretty well behaved. I wouldn't worry about it, and tell your mom that plenty of teen wear jeans to dinner. It's mostly older people that would be shunned if they wore jeans. I don't think they except teens to wear old people slacks or anything =p