One overnight in London

Train from Gatwick ends at London Bridge so all of Southwark should be fine. Close to Tower, Borough market, and South Bank.

Then get a hotel within walking distance of Victoria station. That is where the fast trains to Gatwick start and end. Lots of hotels in that area, I am sure others will list some.
Don't use a bus from Gatwick into London, much slower than the train.

link in #1 isn't helpful, and most Thameslink trains don't end at London Bridge.

There are trains to Victoria, London Bridge and Blackfriars from Gatwick, all 3 stations work for you, you'll have to decide what you want to see then the station choice will be clear.

I did this once and stayed in Victoria area. I don’t remember where (just found it on booking.com). But Victoria train goes to Gatwick and it is a good location for seeing things. My colleague and I walked everywhere. I am returning next year with my husband but definitely enjoyed my short visit.

Beth

Do you need a hotel recommendation? There are Premier Inns all over central London and we like the, for short stays as they are relatively economical. Some have family rooms.

As Tom noted, there are trains from Gatwick to Victoria Station ( Gatwick Express) and to London Bridge and Blackfriars stations ( Thameslink). London Bridge and Blackfriars are fairly close to one another, but London Bridge station is south of the Thames and Blackfriars is on the north side. Any of these three would be a good choice for your hotel, depending on what you most want to see.

Near Victoria Station, you have Buckingham Palace. You can walk from there through beautiful St. James Park to the Horseguards Parade, past the entrance to Downing Street to the Churchill War rooms, and down to the river area to find Westminster Abbey, Parliament buildings, and the tower holding Big Ben, which is covered up for restoration. Cross Westminster Bridge to reach the London Eye.

On a walk the next day you could walk up the Mall to Trafalgar Square and press on to Covent Garden.

From either Blackfriars or London bridge area, you can easily access St. Paul's cathedral, the Millennium Bridge, Tate Modern, Shakespeare's Globe theatre, the Tower of London and Tower Bridge. The pedestrian-only Thames Path links these sites and you walk right along the river with great views of the wonderful architecture, new and old.

And of course many other sites can be visited by tube or bus (we prefer the latter). It's all good.

One thing I would suggest is that with such a short visit, you not devote any of your precious time to watching the Changing of the Guard at Buckingham Palace. It is very crowded, and to see anything of the actual ceremony you must get there very early to claim a spot toward the front, and then wait. You can hear the music and get a sense of the ceremony by walking in the area but away from the Palace. Perhaps someone who is familiar with the timing of the procession can help you figure out the best way to do that if you are interested.

Blackfriars is on the north side [of the Thames]

Well, it's really OVER the Thames, so you won't have any trouble finding it the next day! Note that there are group save fares for families using the regular trains, good value off peak. But stay within walking distance of the train station to simplify things, definitely no tube ride from the London station.

Here is the link to the Gatwick Express Group Save https://www.gatwickexpress.com/offers-and-discounts/discounts/groupsave. This is a good deal and the train is fast. Pull up a map for the tube (TFL) and you’ll see Victoria station center (find the Victoria line and you’ll see it.) We stayed at the Washington Mayfair when we were on the RS London tour. It’s fine, location is great. Nearest tube is Green Park, a few blocks away. Green Park station is one stop north of Victoria Station on the Victoria line.