My catsitter is $78 plus gratuity for a 9 hour overnight stay. Versus $26 plus gratuity for a half hour visit.
While child care and pet sitting are not the same thing, we really aren't discussing "child" care with a 14 year old. And my petsitter probably has more responsibilities when I am out of town than your "babysitter" does - she does the food and water, the litter boxes, and sweeps up the litter. Technically she has other housesitting tasks included in her rate (vacuum, water plants, bring in mail, take out trash, strip and wash the bed she used), but I have a housekeeper who comes to the house as well.
I have seen higher (and lower) rates for overnight catsitters, but it is never remotely the same as their hourly rate multiplied. This is not the same amount of work at all.
I would offer $100 a night, but feel comfortable with $75 a night. At 13 nights, he will make $975-1300 primarily while sleeping and playing video games. This is a nice addition to take off to college with him. I agree with the other poster that he is basically giving up his familiar surroundings and a social life for two weeks, but I don't agree that this is something one is paid a lot for, because it is typically something one does to make a little extra cash when they are NOT planning on doing much more than hanging out and sleeping anyway. If a crisis happened while he was there, I would obviously tip him heavily in appreciation for his help. But I would not do $50 a night, because that is $650, and I would be uncomfortable paying such a low amount after two weeks of work, however relaxed it may be.
#BidenHarris2020 “It’s easier to be a parent this morning. It’s easier to be a dad. It’s easier to tell your kids character matters. It matters. Telling the truth matters. Being a good person matters.”