We took the children last year and loved it - we stayed in an apartment in hte Born area - very trendy boutiquey, wine barish area! There are lots of restaurants/tapas bars outdoors in little squares which meant the children could run around and play a little while we were eating - with all the Spanish children too which was sweet:)
Things we did were:
The Bouqueria market - wonderful tapas bar and the children really enjoyed shopping for bits and pieces there.
The Maritime museum - fascinating!
Open topped bus tour- lovely way to see a fair bit of the city with hop off and hop on facility.
The aquarium as others have said.
La Balconetta - beachy area - with fabby restaurants right on the beach front- we ate at a lovely Morrocan restuarant which had loads and loads of curtained off beds to lounge on while eating!
Rialto park - the one with the boating pond mentioned above.
Las Ramblas - again mentioned above- the children loved the little pet shops all the way along here - although I was a little concerned for the poor animals!
Just before the walkway that goes across to the aquarium there is a little funfair - that was always the first place we went to in the evening so the children could work off a bit of energy before dinner.
I would say to stay as close as you can to the centre - that way everything is within walking distance in terms of restuarants and the main tourist attractions - means that the little ones get far less tired out!
We insisted on siestas for everyone between 2 and 5 each day which meant everyone was well rested before our long evenings!
Have fun - my children still rate it as one of their best hoidays ever:)