If the PI at County Hall is booked (or pricey) have a look at the PI in Knaresborough Place. It's two minutes' walk from Earl's Court station - that station about 35 minutes from LHR on the Piccadilly line - and quite a decent hotel overall.
Not far from that PI is base2stay, a firm favourite of mine and also worth a look. If you'd rather stay a bit more central, check out the Morgan Hotel in Bloomsbury Street (Underground from LHR to King's Cross then taxi to the hotel) or the Park Inn in Southampton Row.
Using PAYG on Oyster will allow you to take the Tube from LHR to central London and ensure you pay the lowest fares going. Touch in and touch out on the Underground, DLR and Overground but touch in only on the buses.
Routemasters run on the number 9 from the Royal Albert Hall to Aldwych and from Trafalgar Square to the Tower. There are six on each sector so if you really want to ride one you may have to wait for a while, depending on traffic.
If you want to take your son to see the lights in Oxford Street and want the view from the upper deck of a bus, bear in mind that because of road works the eastbound lane of Oxford Street is closed until some time in December, and eastbound buses are on diversion from Great Titchfield Street.