South London there is -
Battersea (loads of development of flats etc going in there) - on river opposite Chelsea
Clapham - some lovely streets
Wimbledon - inc. Wimbledon Village which is super posh!
Southfields (on the edge of Wimbledon, but cheaper).
SE London -
I don't know this area so well, but there is Greenwich which is in the river and like a village and you may have heard of the Greenwich Meridian!
I don't know if you know London at all, but few families live in what's called the "West End" which is what tourists usually see - Piccadilly Circus, Covent Garden, etc. Its fantastic, but maybe a bit too intense - not to mention expensive!
In NE London, there is Islington which has loads of shops and all sorts.
Also (more North London), there is Camden which again, is a centre in itself, and famous for its market.
Basically, London is like a cluster of villages that have all merged through history, but all have their own High Streets and different atmospheres. It's hard to sum up.
In your situation with a child, I would suggest Chelsea, Barnes, Putney, Parsons Green / Fulham or Hampstead - if you can find anything in your budget. Your DH can get to Liverpool St from any of these by tube and/ or train. Having lived all over London, there are loads of very nice 'pockets', but I would say these places are a bit less 'rough around the edges' than the nice parts of East or North London (no offence to anyone in North or East London)!