@Sasha - we stayed in the Georgian House Hotel a few years ago. Friendly and cheap but not quite as charming as the pix on their website made out. They may have upgraded since then. It was definitely a good experience, though. And you cannot beat the location.
Sheena - Another option is to stay a bit further out of town. We just got back from London a few weeks ago. We'd decided to stay at the Marble Arch Gloucester Hotel - it's a few blocks from the Baker Street tube stop, in Marylebone, and we had stayed there twice before - but this year, rates were higher than we wanted to pay (even though they are still very reasonable).
Since this trip we wanted to go CHEAP, we looked further out of town and ended up staying at the Holiday Inn Express in North Acton. It's about 15 minutes from the center of the city on the Northern Line of the tube. Right on the border of zones 1 and 2. Easy peasy to get to.
We used booking.com (I think!) and got a rate of 100 bucks a night - wow! There is a wonderful FREE breakfast buffet, which saved us a lot of money - and a Tesco Express across the street, which similarly saved us a bunch.
The hotel seems to cater to families - there were lots of groups of people including adults in a range of ages, plus kids (or not). In other words, I would bet you could get a room for the three of you. It is spotlessly clean and has fresh, modern furnishings.
You also might check on the aforementioned Marble Arch Gloucester Hotel - they have a main building with regular hotel rooms and an annex in an adjacent building, which features big rooms that have kitchenettes. Plenty of room for three people in one of those.
They are frequently running deals on their own website - and are amenable to a call to negotiate prices. We would normally have stayed there, but as above, this trip was all about saving money in London.