How do I convince my mom that this is right?

You probably don't want to mix money and friends together. I wouldn't sell something that was given to me, it's rude. I know the money would be nice, and that you're mother's reasons "sentimental, had it for a long time" aren't very good, but I honestly believe you should give it to your cousin free, or it's possible that she'll let you sell it on the internet, though I know you may not want to have to go to all the trouble.