Impress Father: Get girls phone number with trick.
Impress Mother: Get paid for magic.
Impress Siblings: (Brothers) see "Father" (Sisters) Be the life of a party. Or make an academic correlation to card tricks.
(P.S. I know this sin't exactly what you asked for, but it worked for me to convince them it wasn't a waste of time.)
Learn to pick your battles. Just like you don't want to perform for an audience that doesn't want to see your magic, you don't want to try to convince your family that magic is worthwhile through magic. Just don't bother.
Your best bet is to soldier through the tough parts and just keep advancing your magic. Eventually you'll get good enough to either get paid or do some shows and show them that it's important to you. Otherwise, oh well. Not everyone gets the support of their family. That's sad, but it's true.
Because magic is an unusual hobby, your family probably doesn't fully realize how important magic is to you. That's how it was for me. If you keep at it, you're family will come around, I promise. But you MUST keep practicing. Keep it important to you, and through time, you're family will be totally supportive. Also, magic can be a lucrative hobby, so after a few years, you can make a little bit with it. That effort will show your family that it isn't a waste of your time and resources.
What tricks do you know? I doubt your second statement is true. If you can do a trick well you can probably make some money off of it, if you have the best version of a trick in your area, or the only version of a trick in your area, you can definitely make money off of it.
"People never care how much you know until you know how much you care."
If they can see that this is something you truly love, then they will probably give you the support you want. If that isn't enough for them, then I wouldn't want their support, because if they aren't even supportive of you even if you really love it, then they probably won't ever be fully convinced it's worthwhile. As long as it doesn't interfere with school and other things, in which case I would understand their reluctancy to be convinced it's a good thing.
Lol I noticed no one actually have a suggestion for a trick, one of my personal favorites to do to a tough audience is the invisible deck, however if you don't have the cash to buy one, another great trick is David Blaine's two card monte. Have fun!
Well, i have no idea how much magic your family has seen, but i think you need more than just a trick to change your family's opinion of magic. Possibly get some gigs and bring home some money. Or do some work at a hospital for some kids or something.
But for a trick, invisible deck as mentioned above, or extreme burn