We go to London all the time and LOVE it.
Firstly- for a hotel. Look at Lastminute top secret hotels- 4/5* (really nice ones) for really cheap- rooms for like £100 a night when they would be more like double the amount and more. The cheapest you really going to get in central London is about 70/80 for a 2* hotel thats a bit of a dive- the top secret hotels are really your best bet. To name a few we have stayed at the Grange St Pauls, Grange City, Grange Tower Bridge, Hilton in Holland Park, Hilton London Bridge, The Grosvenor and The Tower- look at the websites and you will see the standard of hotel you get for your money. It doesn't reveal which hotel until you book- but I guarantee it will be lovely. A few we have stayed in have a free swimming pool for guests which is obviously great for the kids. Only downside is breakfast isn't included- but they do let you pay for it and sometimes offer a discount. We went not the weekend just gone but weekend before for a night- stayed at the Grange St Pauls- gorgeous room (twin rooms are two king sized beds so you could get away booking a twin room- you and OH on one, kids on the other- i fitted my three eldest on one bed and youngest slept in cot provided by the hotel) free swimming pool, £10 each for me and DH for breakfast and £7 for each of the kids. McDonalds are in abundance in London- and lets face it, their pancakes are pretty fantastic. You don't want to stay outside central- any money you save your going to pay back in travel.
Secondly- if your going to travel by train, book through Virgin Trains. We have always used the Trainline- booking fee of £3.50 and credit card fee of £4. Thats right, nearly £10 on top of what you are paying for the trains. We looked at virgin this time- no difference in price plus no booking fee and only a credit card fee of £3. Red spotted hanky is a good one too. You may want to look into a family rail card- here in Liverpool we can also use it on the local (Merseyrail) trains too.
Travel in London- get vistior oysters for you (and your partner?). Not only is it cheaper than getting day tickets- it means both of your kids can travel for FREE on the Overground, bus and tube. In tearms for yourself- the bus is the cheapest way to travel. But if you have never been before (or rarely go) then stick to the tube. Everywhere you want to go will tell you its nearest tube station and it announces it on the tube too.
Places to go:
These are some of the places I can recommend
1.Ripleys believe it or not.
2. V+A Museum of Childhood (although this is not central- its out in Bethnal Green)
3. National history and science museum (right by each other which is great!)
4. Ride on the London eye
5. Aquarium
6. Madame Tussauds
7. Imperial war museum
8. Pottery Cafe is great- one in Richmond which is quite far out and one on Fulham Road which isn't exactly a "touristy" area so maybe not the best idea.
9. Obviously a trip to Hamleys has to be in store! Diana memorial playground is nice if the weather is OK.
Maybe you could get the train out to Windsor and go to LEGOland? You can get the train from Paddington or Waterloo- takes less than an hour and there is a shuttle bus that takes you straight there. Does your older one like Harry Potter? You can get the train out to that too I imagine- we are going to look it into it for when we next go down
In tearms of a show if you want to go to one- Matilda is the best. Daniel didn't know what was going on really but the other three enjoyed it. Other good ones are Wicked, Lion King and Shrek- not sure about age suitability on the others as Matilda was the first time we had ever taken them- probably the best (apart from maybe Phantom) theatrical production I have seen.
Eating- Can recommend Carluccio's italian. Best to stick to chains- remember to download voucher code apps to your phone for money off.
Also with your three year old- don't know how he is with walking? Esther got really tired when we were there and LUCKILY I had remember to bring my ERGO so rode in the pram alot of the pram- but it could be because we are just cruel and make them walk rather than pay for the tube. Maybe something you want to consider- your going to be out several hours a day if you want to make the most of things.
Hope this helps!