There's something else that can interest you:
TERRASSA: medium-size city at 40' from Barcelona by commuting train (every 10-20'). Check schedules here, line Barcelona-Vallès, departure Pl. Catalunya, destination Terrassa-Rambla.
On the 6th they celebrate the festival of Culturassa (very loosely translated as "big heritage"). A full day of street activities some of which are deeply rooted in the heritage of the local culture here in Catalonia. Among the different activities, there are castellers, diables or ball de bastons among other. These festival are family-oriented.
It's celebrated in the Parc de Sant Jordi, barely 10' on foot from the train station (Terrassa-Rambla) on the FGC train line from Barcelona. Needless to say, heritage festivals here in Catalonia are free activities, no tickets, no rsvs... just show up.
Touristically-wise, the city has also several sites of interest. Incidentally, one of them, Masia Freixa, is located in Parc de Sant Jordi.
HOWEVER.....
if this is your first time in Catalonia, and its capital city Barcelona, investing too much time going elsewhere outside the city, no matter how "interesting", will prevent you from visiting the city as you should/could given that it has plenty of stuff to fill your three days and more otherwise. Going to Terrassa, while your 10yo will indeed have a lot of fun, will deduct most of a day in all to your available time in Barcelona.
The journey to Terrassa, as mentioned, is short, but the festival lasts the whole day and I seriously doubt your kid will agree to "just see castellers" as some of the most thrilling activities happen at the latter part of the festival, which finishes a 9:30pm, among which Cloenda de Foc ("fire closing") with spectacular fireruns. That's something to experience!
Enjoy!
PS. You don't say where you come from, hence I am not sure what sort of schedules you have back home. Note however that ours are probably different than yours and that on a festival day is normal for kids to stay late with parents, hence 9:30pm is not "late".