I'd guess (no personal experience) that prebooked transportation into the city might be less expensive than a black cab, and for four people it might be a reasonable alternative.
If you want to consider public transportation, I think the Heathrow Express (a train, not a subway) will be running as usual between the airport and Paddington Station downtown, generally every 15 minutes. Paddington Station is less than one mile from the Marylebone tube station. Depending on where your hotel is located, the distance could be quite a bit less than a mile. So you could take the Heathrow Express into Paddington and either walk or get a taxi from there. (Expect taxis to be a bit unhappy about such a short trip unless the minimum taxi charge is quite high.)
Unfortunately, the best current price for October 5 tickets is £16.50 per adult. It appears (I could be wrong) that children no older than 15 are free. That's the price for 14-day advance-purchase tickets, which have to be used on the specified date, though they are good at any time that day.
Non-date-limited tickets appear to be £22 - £25 per adult, so that's what you'd pay if you bought walk-up tickets upon arrival at Heathrow.
A cheaper but less convenient option would be the National Express bus, which goes to the Earl's Court area and would leave you with a costlier taxi fare across town. The NE buses are scheduled to take 30 to 55 minutes. Their frequency varies. Fare for October 5 is currently £6 per person. (Don't know about children.)