Overnight at Heathrow - London

Don't know your definition of inexpensive. Heathrow Cottages BnB. Jury's Inn and Premiere Inn have Heathrow locales.
Yotel at the airport is also an option. Check reviews on the Tripadvisor website for all of them.

Marsha, After renting a flat in London, we stayed in a hotel our last night before leaving from Heathrow last week. We stayed at the Holiday Inn on the corner of Bath and Sipson, literally across the street from the airport. The rooms were standard and comfortable (especially with a return to air conditioning that is few and far between in Europe) We paid a little under 70 pounds for a double room ( two twin beds). I thought it was a pretty good deal, and was worth it for the proximity. Good luck in your search!

I stayed at Jury's Inn last October...it was nice and convenient to Heathrow...use the Hopa Bus Serivce from the door of the hotel to the airport

Marsha, there are a couple of top-end hotels actually located in airport terminals. But you will way a large premium. And, if you stay, sway, at the Sofitel at Terminal 5 and need to fly rom Terminal 3 it's going to take 20-30 minutes to et their, anyway. Heathrow is huge and sprawling and messy. Shuttle into London? It's an hour away by road and Tube. You have your choices of buses, taxis, private cars, limos, the Tube, and two trains. The Tube is the cheapest. Much discussion on here about the pros and cons of taking the Tube into the city from Heathrow. Buses are pretty cheap. Privates cars (driver meets you, takes you where you are going) and taxis will likely run in the 35-60 pound range. At 18 pounds or so, the Heathrow Express train takes you nonstop to Paddington Station in 15 minutes, and it is designed for airline travelers. The Heathrow Connect is cheaper, but makes a few stops. For me, the Heathrow Express is the best deal.

We stayed at the Ibis at Heathrow this weekend. Small but serviceable room. Booked early and got it for 37.00 pounds per night. Great breakfast for 7.50 pounds. Take the red 105 or 111 bus from Heathrow (walk out the National Express waiting room towards the toilet and go right to the bus stand) get on the bus and look to your left. The hotel is about a 15 min ride and it is free. You can tell the driver where you wish to get off and hopefully, he/she will remember to let you off. The Hoppa bus goes to Ibis also, but charges 4.50 each person.
The hotel has no free wifi which I feel it should, since it is an Ibis and not an Etap. There is a MacDonalds about a 3 block walk from the Ibis which has free wifi. Walk out of the Ibis and turn right. The Ibis does have a computer that costs 1 pound for 20 minutes and you can print your boarding ticket for free.

We did the same two weeks ago. We spent our last night at the Premier Inn - Heathrow (Bath Road). I highly recommend it. I paid only 39GBP. The hotel was very clean and had a nice lobby with restaurant/bar. The service was extremely friendly. We took the Hoppa Hotel Shuttle for 4.50GBP per adult (kids free) to the airport. It's about a 15 minute ride.