Generally the approach is your son will call you from Monson and let you know when he expects to arrive at Baxter. He will most likely be hiking with a group of other thru hikers and they have a very good idea on their pace. Given the potential for BSP to be enforcing the 12 thruhiker limit at the Birches campsite (for long distance AT hikers only) quite strictly this year, there is a lot to be said for trying to get some last minute reservations in the park instead of having him rely on the Birches. You can always give him a ride to the birches to meet up with his buds if you are camping nearby in the park. The park keeps a portion of the campgrounds open for last minute reservations by Maine residents two weeks in advance. If a Maine resident doesn't grab them, they are open to anyone but they go quickly.
If you just want to pick him up, you can agree on a time he will be down off the mountain and just show up at the gate and pick him up at Katahdin Stream. There is no restrictions on how many vehicles can enter the park but there are restrictions on parking at Katahdin stream so if you elect to pick him up you have a choice, get a day use pass at KSC which is through the park website or just drive to KSC and meet him at the agreed upon time. The key is if you elect to wing it you can not double park or hang out at KSC and have a picnic and there is no close by place to park nearby. Note that the Day use passes are usually available until the day before during the weekdays but in late summer, they fill up on Friday Sat and Sunday. A real nice alternative is to meet your son at Roaring Brook so he can go over the east side of the mountain if he hasn't been there. Oddly enough there is fairly good cell service (or at least text) on the summit of the mountain so he can call you from up top to confirm his meet time with you. Once you get past Labor day weekend the use drops down in the park but weekend day hikers will still grab the parking passes.
One thing to keep in mind is that if you are staying anywhere in the park, you do not need a day use pass at KSC or roaring brook but you have to be parked before the day use folks drive in for the day. This can mean a very every morning as its 1.5 to 2 hours to drive from KSC to roaring brook and you need to do it before the Togue pond gate is open. Therefore although there may be campsites up in the north end of the park, its just not practical to get to a trailhead in the AM.
The other thing to consider is that it may be crappy weather the day your son arrives, there is a lot to be said for booking a site for a couple of nights so he has the option of finishing on a nice day. Most thru hikers want to finish with their friends.
There are a couple of commercial campgrounds and cabins outside the park with amenities. Many folks elect to stay there and then drive to the park. They may or may not have cell service. Even from the closest one, its 2 hour trip to KSC when you factor in the stop at the gate. By the way unless you have Maine plates, you have to pay a road fee to drive in the park. I don't know the current cost but I think its around $15. If you buy a day use parking pass that will also cost you.
Contrary to the complaints on the reservations system, it is quite straightforward. Just log on the BSP site and do a dry run or call the folks at the headquarters as they can be quite helpful.
The closest place to jump in a lake after his hike is Abol Pond swimming area or the lake just past the main gate where there is a swimming area. The best restaurant by far north of Bangor is the River Drivers Restaurant at the Twin Pines Campground off the road back to Millinocket. Not cheap but great food.