If the weather is dry, Sunset to the top of the last pitch is a good outing. A handline is a good idea and depending on size/ability you might want a short ladder for some of the climbs.
As suggested above, Valley Entrance to Kingsdale master cave and on to Swintso Aven makes a reasonable trip, you can do the passage past toyland as well for extra crawling.
Birkwith is quite a short trip but fun if you take an inflatable but in that area I would favour Browgill as IMO it has more variety and there is also the upstream part of Calf holes as an extra option. Also Redmoss is a good trip but a fair walk and would probably need gear on the climb.
Skirwith is worthwhile although quite a lot of crawling above the waterfall climb, you don't have to do the wet bit at the end.
It does depend on confidence/ability, obviously, but my 13 year-old has done most of the above and I would consider any of them suitable for him.