Like Marc says, us Londoners will probably never have stayed in a London Travelodge
For areas:
Tower bridge - obviously near the beautiful Tower Bridge! Which is great to walk over and take in the sights of London, or walk a bit further up and walk over London Bridge to see Tower Bridge in all it's glory. As far as I know the area hasn't got that much to see and do. Tower Gateway train station can take you pretty much anywhere though. You're only a short stroll from Brick Lane / Whitechapel which is great if you fancy a curry. Head south over Tower Bridge and you'll be in London Bridge area which has quite a lot to see and do i.e London Dungeons
City Road - Busy busy busy (especially if it's actually on City Road). You'll be near Old Street which is good for ventures into Shoreditch, and also Islington for Angel, Highbury etc (good shops, parks, the canal etc). Not really many sightseeing things round here buy very central and easy to hop on the Northern line which will take you anywhere.
Southwark - Google shows me this is more near Southark tube station than Southwark bridge... there's nothing round there really but only a short walk to the bridge and the Thames where you can find all the touristy things such as London Eye, Tate modern, museums, Houses of Parliament blah blah. You should definitely walk the Thames River path (if you and your family don't mind a bit of a walk) it runs the entire way along the Thames from East to West (can't remember exactly which bridge). Don't bother getting the tube about, everywhere is really easy to get to.
Ummm if I had a family and had to choose... I'd probably say either Southwark or Tower Bridge, just because they're nearer the tourist attractions. City Road is in the middle of town too but you'd have to get a bus/tube to see any landmarks.
Hope that helps!
