My 7 year old has a Hudl2, the nice thing with these is unlike my nexus 7, you can add extra storage via the sd card slot, so you can store films etc on the sd card saving the internal memory.
The nexus 7 is fine, but will be quite a bit more money as you'd want at least a 32Gb one, so price wiase, you can't go far wrong with a hudl for kids... i'm thinking I might get one to replace my aging nexus 7 at some point.
Also with the Hudl2 it's parental controls are quite good, you can set it up with yourself as the main account & then add your child their own locked down account, the child accounts have a child friendly browser, where they need you to enter a PIN to allow access. (I've not tried an iplayer app on it, but can do if it something you'd like confirmation on)
This does get a bit tedious when they are constantly wanting access to different websites!
They will also need you to log in to download any apps & then allow it onto their user account. So it's all pretty locked down if needs be.
Most of this can be done on the likes of the nexus 7, but I've not seen an internet browser app that works as good (not saying there isn't one out there somewhere)
The annoying thing I found with the parental control accounts, is there are certain things it won't allow you to let them have, thinking back I think youtube was not allowed on the child accounts, this may have changed I've not looked in a while. but that caused me no end of grief when my lad wanted to watch those damn minecraft videos!!.
You might also do well in laying down some rules regarding its use, as, if like my kids, they'll want to play on it 24/7!! and it's hard working taking it back.