I'm also from the south and almost everyone I grew up with is still in my hometown, and something that wasn't clear to me when I was your age was how important connections are.
Now I'm older, live on the west coast, and I realize connections rule everything around me. While things have turned out pretty well, a lot would have been even easier if I had made connections at your age.
To give you an idea of the value here: If my current startup failed, I could call any one of a dozen people and have a job tomorrow. This kind of safety net is among the most valuable things I "own".
In addition to connections to jobs, you will also develop a lot of connections to expertise. I know and can call up guys who work in hugely diverse fields, from hard drive manufacturing to formal proof verification.
If nurtured, connections also compound just like any other investment. Keep up with the people you meet. You'll get opportunities and meet more people through them. You'll also make some great friends.
Obviously, you don't want to view people purely for the value they can provide you. Good people will see through this anyway. One of the best ways to build really deep industry connections is to try to become the guy that people value as having as their connection. This is where developing yourself comes in to play.
Another tip:
You're probably pretty bright, probably brighter than many of your peers, but almost no amount of precociousness can make up for real industry experience where the rubber meets the road. Take whatever level of humility you currently have and triple it.
Everyone has their own value function in life, and it will change over time. My advice is generally to do what feels important. I wrote about this recently: https://sethkim.me/l/thesolutionspace/?postnum=7.
If the internship opportunity seems incredible, it probably feels important to do, right? And I'd imagine if the pay is really good, you'll be able to travel home a handful of times over the summer and spend time with family/friends. But if spending the whole summer with them feels more important than the internship, it probably is. It's your question to ask yourself, and answer honestly.
On the question of growing up too fast: you will most likely have to get a start on the career you think you want at some point, so the internship sounds like a smart move. I say "you think", because that's likely to change as well (sorry to say). Internships are a great way to dip your toe in the water of a number of career paths to start to figure that out.
Life is a great balancing act of doing what you have to do, should do, and want to do. Balance wisely, but don't forget to have fun along the way!
Home will still be there if things go south. This opportunity won't.