Assuming you are in Israel, and that you have only one day (or two), DON'T waste it on transporting to Dahab or anywhere else for that matter- just go to Eilat. You don't have the privilege to spend additional time on border crossing and taxiing another several hours... I am positively sure that Eilat has more stuff to do than Sharm and Dahab combined (and I DO have been in all of them quite a lot).
Diving is better in Egypt, though, but for your limited time it'll have to wait for another time.
All clubs in Eilat shouldn't have any problem with a teen, as long as both of you provide certification and insurance cards. According to local law, you must provide insurance (can be purchased in every dive club), proof of diving within last 6 months (signed logbook). However, your teenager son might be heavily distracted by bikinied girls on the beach- you'll have to keep an eye on him
If you haven't been diving within 6 months you'll be asked to do a "refresh" dive. If both of you are OW divers (or equivalent) you'll need to either chose a guided dive, or find an additional buddy with AOW/MD/DM etc etc to join you, and max depth limit to 20meters.
So, better make 1-2 dives before arriving, and it's always better to have AOW (or higher) card anyway. If you don't have, then it's time to upgrade your certification level
Dive clubs? You can try any of Manta, Aqua Sport, Siam/Deep, Marina Divers, Udive- all of them within walking distance (crossing the street) one from the other. Just choose the one you find more friendly. Myself, I usually go to Deep/Siam, but most of my friends will choose Manta.
Diving in Eilat is EASY. Most dives are accessible from shore, currents are almost negligible as well as surf- almost like a (warmed) swimming pool all year round. Guided dives are NOT mandatory (unless you feel you'd like to have guidance), so if everything is OK you just buddy up, get geared and plunge in the water. Some of the dive clubs mentioned may also arrange transportation to/from the various divesites at prearranged hours (ask)- some of them free of charge (ask again).
Some of the clubs mentioned have also arrangements with very basic accommodation. For example, I think Marina Divers have ownership of Pierre Hotel (google for it)- not luxury, just a simple room with shower and basic breakfast. Other than that, Eilat has a real wide range of hotels, starting from free camping on the beach (which can be nice in spring/autumn but nightmare or rather "daymare" on summer) ending in luxury-thousand-dollars-per-night hotels.