One day and night in London

What day of the week? Have you ever been before? What kind of things do you like, I.e. History, Art. music? Where are you coming from and where are you going to next? We need to know those details to be able to narrow advice for a huge city with loads to do.

And are you flying there from the US or Canada so you'll be jet lagged? Or coming from the Continent or elsewhere in Britain? Will you need to come into the city from Heathrow or Gatwick or will you be arriving by train?

If you'll be leaving London in the morning, I'd suggest spending the night near the airport or station to minimize the risk of missing your flight or train. There are affordable hotels near Heathrow, and presumably Gatwick, and certainly near all the major train stations.

"What should I do and where should I stay?"
What research have you already done for your trip? You can pick up a copy of Rick Steves London Pocket Guidebook, and get an overview of what may interest you for just $12.99. Includes a map of London, which you will need. A good read on the plane on the way over!
https://store.ricksteves.com/shop/p/pocket-london
Order directly from this website. It is a small guide; fits easily in your purse or backpack. Take it along with you on your trip. It's a good instant reference for seeing the opening and closing times of museums and attractions.
It will help you make your own personal "must see" list. The descriptions and photos will help you choose.

modest budget

what does that mean to you? If you give a figure please indicate if that is in Pounds Sterling or a different currency.

You can download Rick's free audio tours and maps for a few city walks and site navigating. The top two are the top two: Westminster Walk and British Museum. Westminster Abbey is an easy add-on to that plan.

Peggy,

You haven't provided much information to go off of, but here is my 2p.

I'll assume you are coming in by air to Heathrow so take the tube (or Heathrow Connect if you want to splurge) to Paddington Station. There are several budget-conscious hotels between Paddington and Hyde Park.

If the weather is nice (a relative term in England), walk to Hyde Park and beyond. It's actually a pretty easy walk all the way to Buckingham Palace from Paddington... If that doesn't fill up your first afternoon/evening, continue walking all the way to the Thames and take a ride on the London Eye. My favorite time of day for the Eye is late evening after the sun goes down.

If you have pretty-near a whole day before leaving London the following day, I would pick out one major attraction such as Westminster Abbey, St Paul's, the Tower of London, or a museum (Natural History, the British Museum, etc.). Or, if you can fit it into the schedule, you can take a tour of Buckingham Palace as it's open to the public in August and September (google Buckingham Place Tour). I believe you must make reservations on-line and you are assigned (get to pick) an entrance time.

It helps to mention your arrival/departure mode and time, as well as your budget for overnight accommodation.

Thanks to you all for your information and inquiries. We are arriving by air to Heathrow from the western U.S. on Tuesday around noon and will depart from Gatwick for Italy on Wednesday at 5:30pm. We would like to keep our hotel cost under 130 USD if possible. The Westminster Walking Tour with Westminster Abby added on sounds like a good choice. It also looks like the Historic London Walking Tour could be added to this if we head out early enough and walk briskly. Would this be a good expenditure of time? I have been to London twice, but for even briefer layovers than this, so it is all new to me.

peggy

On your arrival afternoon I would head to Westminster Abbey, before photo stops of Houses of Parliament and Buckingham Palace.
The next day I would head to the Tower of London on opening, before photo stops of Tower Bridge and Trafalgar Square. You will need to be at Victoria Station around 2pm for the Gatwick Express train.
I would choose a hotel location near Victoria Station, but I think your budget is a little unrealistic.

David has given you some excellent suggestions. Keep in mind that you will have your luggage with you when you arrive in London from the airport. Westminster Abbey does not allow luggage or backpacks inside, and they do not have a storage facility.

From the Westminster Abbey website:
Entrance Security
It is not possible to enter Westminster Abbey, either for worship or as a visitor, with large or bulky items of luggage. Similarly, it is not possible to bring into services bags, rucksacks or suitcases that would not be classed as hand luggage on aircraft. Left luggage facilities are operated at Charing Cross and Victoria stations, both of which are within walking distance of the Abbey. The Dean and Chapter also reserve the right to search smaller bags before granting access to worshippers and visitors.

You will need to store your bags somewhere before walking around. You may be able to drop them at your hotel and then go out sightseeing.

If you are arriving at noon, you'll probably get to your hotel between 2-3 pm. How are you planning to get into the city? Your room might be ready or you can just drop your bag. If you plan to stay near Victoria Station, which will give you easy access to Gatwick Express, people on this site often recommend Luna & Simone or you could try Premier Inn London Victoria. We've only stay at Premier Inn County Hall, which gives easy access to Westminster Abbey and the Westminster tube station. The PI chain isn't high on charm, but they do have good beds, linens and good prices. They tend to send email specials, so if you think they might work, sing up for the email and see if you get any offers.

I think the idea of trying to do Westminster Abbey and Tower of London hits the big spots.