We order car services occasionally for travel to and from SJC or SFO - you can order a limo specifically, but with only 1 person, a town car is fine and won't cost as much. Also, I have to smile at the idea of showing up to college orientation in a limo while carrying your stuff in a backpack :-) If you order a town car, you may very well end up with a limo anyway - the town cars are all busy, so the driver brings a limo :-) It is not a shared service - they just pick you up and you only - that's why it costs more than a shared shuttle.
For airport pickups, you give the company your flight info and they monitor the flight landing time - so if the schedule changes, they aren't just hanging around. The driver will meet you outside of baggage claim and take you directly to the car.
A taxi isn't a bad idea either though. But since this is the kid's first experience travelling alone, it might be worth the peace of mind to know that the transportation is arranged ahead of time.
Either way, a 2pm arrival at Stanford is probably not going to happen.