Quick London layover sight-seeing options. (updated Itinerary)

Stay near the airport hotel it is cheaper and venture out to London
This is one on of the best chains in Europe for cost cleanlinest service:
http://www.accorhotels.com/gb/hotel-1889-ibis-london-gatwick-airport/index.shtml

It is good you are going in July; it does not get dark til 10 over there.

Use this transport from Gatwick national express http://www.nationalexpress.com/en/airports/gatwick-airport.aspx

It runs later than I believe you will have enough energy to be out considering your flight the next day so don't worry

The British are very strict at the airport border even to US passports. They voted to Brexit part of the reason is to control immigration borders. Make sure you are organized and direct when they ask you a question about why you are there and your travel plans. They will ask you for proof of where you are staying and sometimes ask for proof that you have a ticket leaving the next flight.

You got to do a little more research on how to get around using the tube on the day in London so you are not killing yourself walking to the sights you are interested in seeing.
First you should read a guide book or watch Rick Steves London video on youtube.
https://www.youtube.com/watch?v=4p1rzw-7Jho

My motto "only you know what you are really interested in seeing"

"Is customs typically a big issue?"

"Customs" is typically not an issue (unless you're unlucky enough to have your bags searched), immigration on the other hand can be lengthy, depending on the volume of passengers and the amount of border staff on duty.

The abbreviation for Gatwick Airport is LGW.

There are Thameslink commuter trains from Gatwick to Blackfriars (good pub, close to St Pauls and The City, but as Emma says far from the West End and many of the main tourist sights for a new tourist) but it is slower and less frequent than the Gatwick Express and Southern services into Victoria (lots of construction around Victoria).

If you stay at the airport you can drop your bags, freshen up, have as much of the day and evening as you want in London - some trains run all night but after midnight they become less frequent - and you will be on the spot for your flight in the morning.

Another option is to stay near the airport and take the train to Brighton, see the Pavilion and relax a little.

Nigel points out LGW as the correct airport code for London Gatwick.
GAT is a small aerodrome in France - Gap Tallard.

I will definitely check out the airport hotel as I'll feel better being closer to the airport in the morning! I'm booking a room soon. One limitation is that there are four of us traveling (46, 44, 18, 16) and it looks like we'll need to book 2 rooms at that hotel. It is still cheaper! I'm happy to have those long days so we will make the most of the arrival day (July 2nd).

Thanks for the advice on immigration. We'll be direct with our answers. I'll watch that video for certain.

I will check out the Gatwick Express train or Southern trains. I'll also check into going to Victoria as an option. I greatly appreciate it!

Thank you for the correction. My overall flight is ORL (Orlando) to GOT (Gothenburg, Sweden). I had gotten confused with the layover in London Gatwick (LGW).

If you stay at the airport you can drop your bags, freshen up, have as
much of the day and evening as you want in London - some trains run
all night but after midnight they become less frequent - and you will
be on the spot for your flight in the morning.

That sounds like a plan.

Itinerary Update

Thanks to some great advice here, we've further refined our itinerary. We know we are going to be tired, jet-lagged, but are going to make the most of our afternoon/evening. Since we literally will have 6 - 8 hours to see a small portion of London, we want to limit our visit to what we can walk or take a double-decker bus around a bit. I realize it doesn't get dark until 10pm, but we will do what we have energy to do before returning on the train.

1. Arrive in London Gatwick airport at 8:10am (on Sunday, 7/2).
2. Take airport shuttle to "ibis London Gatwick Airport Hotel"; check in bags, get day-bags and head back to the airport on the shuttle. We may take an hour nap...
3. Take Gatwick Express train from the airport to London Victoria Station [Ideally, we'll be here by noon]. This will cost my family of four a little more than $110 round-trip based on May's dates (we can't forecast prices any later on their site).
   
4. Walk to Buckingham Palace and make our way down toward Westminster Abbey, the Houses of Parliament and up to Trafalgar square. What exactly we'll do depends on how easily we can travel by bus only and how much energy we have.

My questions:

1. The Gatwick Express train website asks us "Departure after" time. When I'm ready, should I put the earliest time we'll possibly be at the station? Should we choose the "Anytime Return" option to give us that flexible return option? Can I get these tickets at the airport pretty easily?
2. Is an "oyster" card what we need to each purchase to ride the red double-decker buses?
3. Our flight to Sweden leaves at 09:55 Monday morning. What is the typical time we need to arrive at the airport prior to our flight departure?
4. Are there any "must see" places in the general area I described?

If you like WW2 history the Churchill War rooms is a nice museum, National Gallery for art.

In terms of airport arrival I have always used 2 hours prior to my flight with easyJet. Drop off my checked bag and through security. Always been plenty of time but this is early around 7 AM. The Gatwick website seems to suggest 3 hours.

http://gatwickairport.com/at-the-airport/flying-out/check-in/

Perhaps it's best to check with your airline.

Why aren't you considering taking the Thameslink Train from Gatwick to say, Blackfriar Station?

Believe it would be much cheaper, takes about the same time (an hour) as the Express. You could then walk along the Thames Path for about 20-25 minutes up to Westminster Bridge. Excellent views of Parliament and Big Ben from this direction.

If you like WW2 history the Churchill War rooms is a nice museum, National Gallery for art.
In terms of airport arrival I have always used 2 hours prior to my flight with easyJet. Drop off my checked bag and through security. Always been plenty of time but this is early around 7 AM. The Gatwick website seems to suggest 3 hours.
http://gatwickairport.com/at-the-airport/flying-out/check-in/
Perhaps it's best to check with your airline.

Hopefully, with just having carry-on bags, we'll save a bit of time at the airport. I will check with them and hopefully check in early too with Norwegian airlines. Thanks for suggestions on events.

Why aren't you considering taking the Thameslink Train from Gatwick to say, Blackfriar Station?
Believe it would be much cheaper, takes about the same time (an hour) as the Express.

I didn't know any better, till now. Your suggestion saves me about $50 plus it looks like any other transportation in Zones 1-6 are covered. This does open us up to possibly jumping around a bit more if we choose to do so.

This is what the reservation said:

Off-Peak Day Travelcard £43.80
Valid at off-peak times only. Includes unlimited travel on trains, London Underground, buses, DLR and Tramlink services within the zones specified. Refundable.

Gatwick AirportLondon Travelcard Zones 1-6
Valid only on Southern services.

I'm revisiting this topic now that I can get actual prices since I am closer to departure.

We are supposed to arrive 8:20am on July 2nd. We plan to go to our Ibis airport hotel to check in and then take a train to Blackfriar station. Looking at the ThamesLInk website, ideally, we'd like to have an "open" leave time because of the uncertainty of departure time. We'd also like to be able to have an open return. Using the site, the best price I see is around 79 pounds (it was less than 50 pounds when I checked a few months back).

1. Is there a way to pick an open departure?
2. Do I simply buy tickets when we arrive in London at the station?
3. Can I get "Super Off-Peak Day Return"?

Best option is a Super Off-Peak Day Travelcard which is valid on 2 July as it is a Sunday. Would enable you to travel to and from London on Thameslink and on the buses, tubes etc whilst in London. £12.30.

Best option is a Super Off-Peak Day Travelcard which is valid on 2 July as it is a Sunday. Would enable you to travel to and from London on Thameslink and on the buses, tubes etc whilst in London. £12.30.

How do I get one of those?
When should I get it (hoping I can buy one on arrival)?

I still can't find the same rates quoted. I did find this link: https://www.thameslinkrailway.com/tickets/super-off-peak.

I enter the following:

• Gatwick Airport as departure and London Blackfriars as destination.
• Outbound Sun 02 Jul leaving at 10am
• Return Sun 02 Jul leaving at 21:01pm

This returns several rates. I then click "View all available fares" to find

• Outward: Super Off-Peak Single for £23.00
• Return: Super Off-Peak Single for £23.00

Total £46.00.

Scrolling down further, I find "Super Off-Peak Day Travelcard" for £32.40. This is obviously the best rate (or is it?) Do I buy this now or wait?

Clicking the "?" shows details below:

Valid at off-peak times only. Includes unlimited travel on trains,
London Underground, buses, DLR and Tramlink services within the zones
specified. Refundable.

Gatwick AirportLondon Travelcard Zones 1-6* Valid only for travel on
Thameslink services.

4 Adults at £8.10 GroupSave Discount£32.40 This fare contains a
GroupSave discount and is only valid if all passengers travel
together. See details of exclusions and restrictions. Valid dates

OutwardSun 02 Jul 2017 ReturnSun 02 Jul 2017 Break of journey

OutwardYou can break and resume your journey ReturnYou can break and
resume your journey
*Valid to London Blackfriars City Thameslink London Bridge