Adventurous budget-minded family [London]
if eating on the go is an option, check out some of the markets, where you'll be more easily able to mix and match between different stalls and their specialisms. Broadway Market or Brixton Market would both give a broad range of options as well as the more central Borough Market and Maltby Street Traders
It really depends on what your budget is and how far you are willing to travel.
For me the best value is out of town in the South Indian places in Tooting, Wembley and East Ham. These though are a good half an hour away from central London
More centrally for under a tenner a head I would recommend.
Bonda- Maylasian cafe near Paddington station.
Patogh- Iranian place on Crawford St, just off Edgware Road. Giant breads and grilled meats for about 7 quid.
Indonesian Mini Market in a little shopping arcade at 57 Charing Cross Road.
If you happen to be going to Portobello Road Market then the Morrocan stalls on nearby Golborne Road.
These threads for the best of 2012 <£30 and also <£15 may be of some help. They aren't all London tips but are mainly.
There have been threads on budget dining in London. Here is one:
If you go to the Search feature in the upper right-hand corner of this page and search for "budget" you should be able to find a number of past threads with helpful recommendations.
And, of course, the search facility will also find previous threads on the various "ethnic" restaurants the capital has to offer. Presumably the OP may wish to search out "ethnics" which London does well but are not well done where the OP is.
I suppose the other consideration is how far the OP wishes to travel from Mayfair. Walk? Cab? Tube?
The bargains are often to be found at lunchtime but that doees cut into the touristing. A nosy through Top Table is always going to be useful for the budget eater.