If I was travelling with kids I would think about South Kensington, Bayswater or Bloomsbury. The first area is known for museums and is close to the shops of Knightsbridge. The second area is just to the north of Hyde Park and Kensington Gardens and is good for families whose kids like/need a regular place to romp around. The final area is near the Brish Museum and close to Covent Garden and the sights in the city (St. Pauls etc.)
If cost is not real barrier than the hotels of Mayfair (Park Lane) or Belgravia are perfectly lovely but perhaps a little formal for active kids.
In South Kensington, as examples, you might check out the Kensington Park, the Gloucester or the Harrington Hall. In Bayswater you might check out the Hyde Park Thistle, The Royal Lancaster or the Corus Hyde Park. In Bloomsbury, I stay at the Bonnington. The Bedford and Jurys Great Russell Street are also worthy choices. St. Margarets is a fine B&B which many famies enjoy. Also the Jenkins, another B&B.
Finally, if being smack dab in the center of it all turns you on, check out the Thistle Charing Crossn and get rooms overlooking Trafalger Square. (but beware of maddening traffic)
Just some ideas. All of these places have their friends (and foes) here on the Tripadvisor forums. Check out the comments. Happy trails.