Layover or Stay Overnight?

If it were me, I would choose option 1 and continue on to Prague at 2pm. I always try to avoid one night stays because for the most part, they are time and money wasters.

Ten hours of walking around a big city after a trans-Atlantic flight sounds very bad to me. If you can sleep on planes and arrive at your destination ready to go, maybe it would work for you. But I think most people would find it exhausting. I would prefer to get to my final destination the same day so I could hit the ground running the next morning.

Madrid is a nice place to visit. Not my most favorite place in Spain, but one where a few days are worthwhile. Spanish culture will contrast with the rest of your trip. I might be tempted to spend several days there before going to Prague.
I would not consider option 2 since it's such a short time. I spent a career doing single nights in places and I try to avoid those now. Keep in mind (at least for me) day three of any trip is the long one when things start to catch up a bit.

Option 1 is the best one, IMO.

Option 1 is the best one, IMO, for all the reasons stated above.

The airport is far from the core part of Madrid, the areas you would want to wander and explore. It's a 45-minute train ride if memory serves. So you probably wouldn't want to stay near the airport and train back and forth for just 1 night. Another vote for option 1!

While you should have enough time for option 1 that isn’t actually a very long layover if your second flight is on a different ticket, as it looks like it would be. If there’s any significant delay with your flight into Madrid you could risk missing your flight to Prague, in which case you’d likely have to buy a new ticket at walk up price, and if your Prague ticket is a round trip (do you have to go back to Madrid to go home?) then missing the first leg usually cancels the rest of it too.

Jennifer,
If you are purchasing the flight from Madrid to Prague on a SEPARATE ticket, I would strongly advise you to stay overnight in Madrid or at least, book a flight later in the afternoon, in case your flight to Madrid is delayed.

For me, it would depend on how long the trip is overall. I always recommend this way of choosing destinations. Getting over the ocean is job one. Build the trip around that. If you only have a week, do Option 1. If you have more time, do Option 3, which is stay in Madrid for a few days to get acclimated to the time change with no agenda. Or Option 4, which is Option 1 with Option 3 added on at the end (assuming your return flight is also from Madrid). Unless it's high summer (during which time Madrid will be HOT as the devil), in which case Option 1.

Thanks everyone for the thoughtful replies and suggestions! It appears Option 1 is the clear winner :)

Unfortunately, from what I've read, Iberia (who we will be flying with from the US) does not allow you to combine itineraries, so even if we book an Iberian flight from Madrid to Prague we will need to recheck the luggage and everything.

Appreciate the suggestion that 3.5 hours might not be a long enough layover in Madrid. It looks like there is a flight that leaves Madrid at 4:00 on the day of our arrival which would give us 5.5 hours. Hopefully there is a good lounge at the airport!

Personally I think I’d choose option 3, which is to buy a ticket direct to Prague and redeposit my AA miles for something else, no matter how good the Madrid deal is, given what a hassle and delay, plus additional cost from Madrid to Prague, it is to fly to Spain when your destination is the Czech Republic on the other side of Europe and you have no interest in visiting Spain.

Madrid wasn't on my bucket list of cities to visit either, but we greatly enjoyed spending time there. Two absolutely world-class art museums, beautiful parks and very, very modern. It's so unlike the rest of Spain, and consequently fascinating. So as a result I recommend spending a couple of days there before traveling on to Prague.