I would also go with the Four Points; I've done the walk to the beach from there, and runs down Diagonal (also to the beach / marina), and it's a nice pleasant area to stay in. If you're brining a stroller as well, you'll have more space to maneuver around than you'd have in the old city with the LM.
Rooms aren't spectacular (and don't get the rooms under the gym - had that for a week and people love to drop the barbells), but it should have enough room for your son.