It sounds like you may have already purchased the SFO-LHR tickets in which case, you're probably locked into them.

Also, are you saying that you want to spend the first two days in London and then continue on to Barcelona?

Two days in London isn't very much time really. By the time you allow immigration and customs, getting to hotel etc with a family of five it will be 10am or so . Then you will need to allow at least 3 hours to get back to LHR for the trip to Barcelona. You will not see much in London in the two the days as a lot will be taken up with traveling time. I guess it depends on the age and condition of the family members. Everyone between 12 and 60 probably not a problem. Other ages might take more time for everything. And don't forget the 8 hours time difference and the effects of jet lag. It can be a killer!

Also remember that Easyjet and many of the European carriers have very strict baggage allowances. You will only be allowed one very small carry on, think one purse or one very small laptop, not both, and you may find you'll be paying for excess checked luggage depending on how much everyone packs. So any potential savings could be lost in extra fees, transport for 5 people (will always need two cabs etc) and time.

If you haven't already booked tickets others have made good suggestions. See about buying a thru ticket with a stopover. Or even avoid London and use another European city as a transit.

London has enough to keep someone accupied for a couple of weeks. It is a destination in itself. Plus it is very, very expensive with the current exchange rate.