There is a couple questions you need to ask yourself first before deciding what direction to go...
1. what else are you using this camera for? Do you have a use or anyone in the family that wants to use it after your trip?
2. how much additional money are you willing to spend knowing the answer to number 1?
3. as
@Brians_myth mention, what are you willing to carry with you in addition to everything else?
4. what do you plan to record? does it need to be waterproof?
Mobile phones are a good starting point, you'll see 90% of the people around WDW using the phone. It's in your pocket anyway and your teen will probably have their own. consider what storage ability it has and if you can add a memory card to it, it's a good spend to have more space. If you plan to record a ton, also be ready with how your going to recharge it during the day and how your going to have empty space to record the rest of the trip if you fill it up.
gopro I like as well, especially the new hero5 that is waterproof out of the box. you can use that camera when others are hiding from the rain, not worrying about protecting it on rides like kali, pirates or splash. no worries about footage pool side. I recommend getting the floating handle that gopro also sells as an accessory. it's not a selfie stick and doesn't extend so it's allowed in the parks, it makes it easier to hold on rides and has a wrist strap to ensure you hold onto it during the faster rides
and the best recommendation, start using your phone now.. learn if you liek it, how much video video fits on it before you fill it, get your arm strength up for holding it in the air for 15-20 minutes recording a disney show. try it at night to see if you can even get video of some things you want to record while there. then you can make an educated choice on if your phone is not going to work out but at the same time learn what you like and don't like about using the phone so you are comfortable using it when you get there instead of futzing with it..