In reply to stuartf:
Over the years I've done a lot of holidays with my siblings and their children. Each holiday we do a hill, and the target gets more ambitious as the children get older. Some of the early ones we did were:
High Rigg. Climb it from the church at the N end. A nice bumpy and wild top.
Castle Crag. Great views, big cliffs. Fits the bill of a proper mini mountain.
Binsey. Nice top, but definitely not a 'pointy mountain'.
Either of the Mell Fells.
Ling and/or Sale Fells. Again, not pointy but quite wild and quiet.
Arnison Crag. A bit rocky, and great views down Ullswater.