Does he like chocolate? Take him for some churros con chocolate at La Granja on C/ Banys. You purchase the churros at the Xurreria up the street before going in, or as we did, send someone out once you're seated so they're fresher when your chocolate comes. This would also fit into Lynda's suggestion of walking in the old town and finding bits of Roman walls since there's an actual section of Roman wall a the rear of the dining room.
You could also drop by a Bazar Chino (dollar store), purchase a couple of ping-pong paddles and ball, and go play on the concrete ping-pong tables in Parc de la Ciutadella. It's about the cheapest most fun activity you can do in Barcelona.