Near airport, Premier Inn was pretty good, really. Very comfy bed, but bring your own soaps, etc.
In London for hotel near sights, we found when we searched the Travelodge site over there
http://travelodge.co.uk
there were good deals to be found, both the one in Southwark and the one in Windsor were good, comfy, with TV, no phone, but good buffet. I can't find travelodges on any of the hotel search engines, and the prices are sometimes really low, so they are worth a separate search to see what's on sale for your dates.
With such a short trip, you'll do well to get really specific about what you want to see and do, plan out the minimum , and then add things as you feel like it. It can be tiring.
If you have any possible connections to friends, or any club or church or anything, it might be good to connect with a person over there to help with such a short trip. It can be a little daunting to figure it all out so fast.
If you take the Express train from Heathrow, into the center of London, it goes over to Paddington station I think, so you may want to find a hotel near there. Keep it as simple and low-stress as possible :) This is one of those times that spending some additional money for the best location is very worth it. Depending on the time you're going, there can be dorm rooms (nearly all en suite) for rent in the center of London too. If you book now, the exchange rate is really good for us again, so it can be cheap if you pre-pay.
For your first time out, you may want to check out the YHA youth hostel association
https://www.yha.org.uk/places-to-stay/london
You can often get a private room, and you'll also have some extra guidance with the local area and sometimes more friendly lounge rooms with other travelers. I stayed at Pax Lodge because I'm a Girl Scout volunteer, but it's in Hampstead, not the very closest to the sights you probably want to see.
Good luck, keep it fun as possible, and remember that everyone else has also had their first times visiting that they also messed up badly on :) We've all gone through it. Be willing to hail a cab if you get lost, and enjoy the rest of the time that you WILL get right :)
-Alison