Toddler happiness is considered a public responsibility in Spain: in any public place people around you put every effort into making your toddler happy: whenever he looks boring or is crying, everyone makes their best to entertain or to calm him.
La Pedrera (Casa Mila). At the exit from museum part (same floor as The Pedrera Apartment), La Pedrera hosts a small toys store. Mid-XX century-style metal toys (including musical boxes and whirligigs); fascinating books with pop-up pages (especially check Alice in Wonderland and The Wonderful Wizard of Oz, €30 each). Check also several books How to Speak to Children About Art from several galleries (Tate Gallery, Metropolitan Museum). To get to the shop, you need to buy a standard ticket to La Pedrera.
There is a underground mall somewhere between Placa del Pi and Born area, with a good selection of child shops.