We are connecting in London between two long flights in February. We have no more than 35 hours and instead of our usual, stay at an airport hotel, would very much like to go "somewhere" interesting for a good sleep. For instance
I have entertained close to either the Tate Britian or Tate Modern. We are open...we do not know London so are looking for suggestions for easy access and a comfortable place to stay. What say ye experts?

Firstly, what airport are you having to reach (if it's Gatwick/Heathrow then consider staying near the stations for their respective express train services)?

Secondly, budget?

in addition to m_kingdom's questions -- what times are your arrival/departure? Could make quite a difference re where to stay.

We are arriving at noon on the 6th of Feb. and departing at 4pm the 7th allout of Heathrow...so that 35 hrs is a bit inflated. Actual available hrs outside of the airport is actually more like 24 hrs. I would like to keep it around 400 US $'s. I have considerd staying near the best/quickest train locations but, as I have only flown through London, do not have the insider knowledge of that.
Many thanks for any insights and names/locations you might throw out.

You should be able to find a decent hotels in central London given your schedule and budget--it's an off-peak season in February. Noon arrival at Heathrow is usually not that crowded (vs, say early morning when there are many red-eye long-haul).
You can catch either Heathrow Express or Heathrow connection (slightly cheaper) to Paddington, which is not far from Mayfair on taxi (say, GBP8-10). Or if you don't have much luggage and don't mind slightly longer transfer, take the tube.
Just bare in mind it gets dark VERY early in winter, around 5 or 5:30pm.

There is actually a Hilton attached to Paddington Stn. Dunno as I haven't stayed there, but opened relatively recently, entrance/lobby looked fine. Though Hilton is a Hilton is a Hilton with limited local flavour IMO...

Re-reading your original request, you want "a good sleep", in which case I'd say that staying at Heathrow will guarantee that and remove any hassle associated with the journey into/out of Central London. Also, you will (if you choose the right hotel) have access to a pool/gym which is something few London hotels offer!

Staying at the airport mwould waste valuable sightseeing time, IMO. Becaause of your flight times, you should be able to get to central london by about 2pm to check -in. I suggest a hotel in the centre, near something you would like to see in the morning, before you leave.
You will have the whole afternoon and evening in London, plus the morning up until about 12 noon. Then back to the hotel, [see why I suggest staying near your morning destination] quick change, tube to Heathrow, check-in for 4pm flight. I think this works!