There are many rentals in Ramat Eshkol, right next to Sanhedria.
Look in the local pirsumit (ask your son to pick one up, they are all over the neighborhoods) or look in diradirect.com you can find a rental under $100. a night.
Have you been to the Churva shul yet? it was just rebuilt and open to the public.
Ir David or Migdal David are spectacular at night.
What about the Shul that looks like the Bais Hamikdash (forgot what it is called and also nearby.
If you want cheap eats, but local (most of the restaurants are under Rav Rubin’s hashgacha), go to Rechov Paran nearby. For a real treat for your son, take him to entrecot in har chotzvim (nearby) for a great meat meal – also under Rav Rubin – reservations necessary.
Big question, are you renting a car? If you only have one day with your son, without a car, the trip to Rechovot will take most of the day.
Regarding food, nearby is a fleishig Hadar Geula where you can do takeout for shabbos and any store sells paper goods for Shabbos.
Big question is how long are you going for.
I know when I visited my daughters in seminary, most of the day was spent with my daughter in her classes. It was wonderful for her and for me. I imagine learning with your son could be the highlight of your trip.
FYI, Egged has an english web site, so your can map the busses you will need to use/find BEFORE you go which will make life much easier
Have a wonderful trip.