Legally he can’t drink, not even with a meal. I don’t suppose a waiter would bother asking for ID so it’s up to you if you want to try.
You do seem to have picked a bad weekend for it as it’s pissing down, and forecast to carry on until Monday…
Anyway, a few ideas:
The rooftop bar in the hotel on Plaza España is fun, with some glass floors a long way up for scary entertainment. Not ideal in the rain, but presumably there won’t be any queues!
VOSE films at the cinema (VOSE – original version with subtitles in Spanish). Something you could obviously do in the UK, but given the weather…
The Retiro park on Sunday morning is quite fun, as is the Rastro. Keep an eye on your wallet.
There are loads of restaurants north of Gran Via, and they’re generally a bit less touristy than the ones to the south. Although if you’re hungry (and eat meat) you could try a “cachopo” in Casa Antonio in La Latina. Cheap and cheerful and filling. The cafe at the top of El Corte Ingles next to FNAC is reasonable value and IIRC it has some nice views.
FNAC is probably about the best you’ll get for shopping with a 16 year old. Computers, games, books, that kind of thing.
Parque de Atracciones is an amusement park but I’m not sure how amusing that would be in the rain.
Take a train out to Avila, Toledo or Segovia.