Don't stay in! There's loads to see and do.
Are you sure he won't be interested in a show? Not even Hamilton? Or The Play That Goes Wrong? Or if he likes Ghibli films, Totoro.
You could take him to The Lucky Saint in Marylebone which is an alcohol-free pub.
Or you could go for dinner at Brick Lane, famous for its curry houses, or walk to the top of Brick Lane and go to the all-night bagel shop. That area is full of street art - really good graffiti and murals, tiny sculptures in odd places. Interesting to walk around. It's hipster and looks a bit rundown in places but I've been there loads of times and never had any trouble. And it's not far from London Bridge for a cheap taxi back.
You could go for a cruise down the river. The Uber boats are not too expensive, and iirc some have cafes aboard, or you could pay for a proper cruise. They usually pick up from Tower of London, so very near your hotel, and go up to Westminster and back.
London Eye is good at night.
If he likes swimming, you could swim at the outdoor pool in Covent Garden (Oasis) as the sun goes down. It is open for lane swimming 6-9pm most evenings. If he likes comic books, graphic novels or manga etc you could look around Gosh on Berwick Street or Forbidden Planet then go for an evening swim at Oasis and then have dinner in Chinatown nearby.