Pete. look at it a bit differently, your family won't be allowed to get in the start area to "see you off". At the same time, the transport for runners out to the start is superb from the central London stations, so you'd be better picking a hotel where you can reach the start and they can get to a good spot to watch and then get to meet you at Horseguards after the race.
Cheap hotel? what's your budget? I just looked at Travelodge and I would go for a family room at Aldgate or Bank, you could walk to London Bridge station from either of those and your family will be right on the course to see you-and you'll know where they'll be which is a good thing.
Think about your breakfast, take what you need with you in case your hotel doesn't offer a decent ealry breakfast.
Where are you travelling from, and how? The first year I did the race, with 2 small kids, we stayed at Dartford Holiday Inn, piece of cake to get to, and a Direct train in to the start from Dartford Station, I think it will be less than £100.
Message me if you want any more help.