Stay the night in Cordoba?

Of your options I would do number 3.

But what I would really do is:
Night 1 - Arrive Madrid mid after noon and take a train to Cordoba and spend the night and most of the next day
Nights 2, 3, 4,5 - Seville (so three full days)
Nights 6,7 - Granada (leave for Madrid early afternoon, so you have 2 full days in Granada)
Nights 8,9 - Madrid (one and a half days)

Staying a night in Cordoba is a good idea. It's pleasant in evening after the crowds thin out. But it does mean another change of hotel in just 9 nights. And you cannot do everything in one trip. Perhaps you'll return?

There is an option 4. The journeys from Madrid to Sevilla and from Granada to Madrid both pass through Cordoba so you could break one of those trips and spend most of the day in Cordoba before completing the journey later the same day. Obviously you need to buy appropriate railway tickets and store luggage at Cordoba.

Of your three current options, I least like number 1. You only have two full days plus two part days in Sevilla and I think you need all that (& more) to see the city itself. If you want to daytrip to Cordoba from Sevilla, then add a night in Sevilla.

Although I prefer Granada to Madrid, I'd take day away from Granada in this case. You might be jet-lagged on days 1/2 and on day 9 you have to get to Madrid so don't get a full day. There is enough to see in Madrid to fill your already limited time without taking away more. Alternatively, you could push onto Cordoba on your arrival day - but that would be a long trip assuming you're arriving from across the Atlantic.

How about putting al. your nights in Madrid at the end and take train to Córdoba on arrival day? That is more efficient than dividing your time in Madrid. Then you could have two nights in Madrid without much loss.

To Dream the Impossible Dream!
How to ideally spend nine nights in the Garden of Spain.

I am presuming this is your first Spanish trip, you a flying non-stop from an eastern U S point of departure, you are relying on public transportation once you arrive and you are in the process of making arrangements for lodging and transportation.
Using your stated itinerary, days 1,3,6,9 & 10 are travel days - some a considerable distance.

You have heard "good things about Cordoba" - and every one of them is true. Have you heard "good things about Toledo?" Every one of them is true, too. And Ronda, and Segovia, and, of course, Barcelona and the Basque region......this list goes on.
Some people love Italy, some love France, some love Germany.....all for good reasons and all those reasons are personal. Some places just make you feel like this is your destiny. Spain is like that for me......in fact, I'll be there in late January.....Madrid south to Andalucia.....and it's not my first fiesta.
Maybe Spain will be that destiny for you....maybe it works its magic on you like it did on me. Maybe it will be just another wonderful European adventure for you - but the search for your dreams will lead you elsewhere. For me, no amount of time in Spain is sufficient. Do I recommend you somehow add Cordoba to your plan? I recommend you prepare yourself for a dream come true.

You've gotten some good ideas here. The one I like best is going straight to Cordoba on your arrival day. It means no extra hotel changes and not splitting your Madrid time, which should make you just a bit more efficient in that city. It will probably be best to wait until you get to Madrid to buy the train ticket to Cordoba. I assume you can get that ticket, along with the train ticket to Atocha Station, at the airport train station. That's important. You do not want to have to buy the Cordoba ticket at the staffed counters at Atocha (assuming you don't want to spend hours in line, trying to stay awake). Or perhaps Renfe has a smartphone app that you could use to buy your ticket as soon as your plane lands.

It looks as if full-fare AVE tickets cost about 63 euros. [edited] There are a few AVCity trains that are about 20 euros cheaper and take 15 minutes longer. Trains are fairly frequent, which is a good thing, because they do sometimes sell out. I see sold-out trains for tomorrow.

Atocha Station is large and a bit confusing (even if you aren't right off a red-eye flight). Allow a bit of extra time between the local train from the airport and the departure for Cordoba. There are lots of places where you can pick up food for the trip.

The main issue I see is that Madrid is split. I’d head to Seville on arrival, and do all your nights in Madrid at the end. Breaking up a visit to a city makes it harder to get the feel of it, leads to too much hopping. And ENDING in the city you fly home from is easiest. Spend the jet lag first day in transit, in this case on the train to Seville.

I personally would add a day to Seville, taking it from Madrid, but do Córdoba as a day trip. I dislike 1 night stays and it is great as a day trip. However you then would need more time in Seville. I am also a huge fan of Granada and less so of Madrid (still enjoy it, just prefer Seville and Granada).

Thanks everyone! These responses have been great.

I didnt even think about taking a train to one of the cities after arriving in Madrid. How hard is it to get a train ticket the day of? I wouldnt want to do it in advance, as there could be flight issues so I wouldnt want to be tied down. We are flying from the USA to Munich, and then Munich to Madrid (arriving at 2:30).

I'm leaning towards adding an extra night to Seville, and possibly making Cordoba a day trip. I just dont want to change hotels too often, and from what I read, it seems like my wife and I would enjoy Seville/Granada more than Madrid (although we would like to explore Madrid a little before we left).

You can visit renfe.com periodically (probably best on the same day of the week you'd be making the trip) to see what the ticket situation looks like. You should do that very early in the morning our time, since Spain is 6 hours ahead of us. I do see sold-out trains sometimes, so I am not going to promise you that you'll be able to get a ticket on the first train you have time to catch. Maybe yes, maybe no.

We took the train from Seville to Cordoba in the early morning and had that day and one night there, which we felt was enough. We stayed in Madrid for three nights but spent one day doing a day trip to Toledo, and wish we'd had more time in Madrid as it was a beautiful city with so much to see.

Tickets for the Mezquita in Cordoba aren't sold in advance so we booked a tour through GetYourGuide before we left home and enjoyed it. The Mezquita was beautiful. Make sure to book tickets for the Alhambra far in advance as they sell out quickly. We had a day and a half and two nights in Granada and thought that was enough. We were able to see all of the major sights. We had three nights and 2 and a half days in Seville and loved it. We visited all of the major sights and had time to wander through Triana, across the river. The Placa de Espana there was one of our favorite sights - incredibly beautiful.

Are you traveling by train? We are over 65 and bought a Dorada (gold) card after we arrived (6 Euros each), which saved 25-50% on tickets. The trains were marvelous - always on time! We booked the fast train from Barcelona to Granada in advance, and when we arrived there, I just gave a clerk a list of cities (from-to) with the times we wanted to book and bought all of our tickets for the trip there. We enjoyed Spain even more than we had anticipated! People were friendly, hotels were inexpensive, and food (we mostly ate tapas) was good cheap!

Thanks! Yeah, we plan to travel by train, although I know some routes the bus could be better. We'll explore our options once we get closer to the date.

I would love to spend more time in Madrid, but I dont want to overextend ourselves. I'm hoping if we love Spain enough, we can come back and split another 9-10 day trip between Madrid and Barcelona. I just dont want to be on trains the entire time. As of now, our itinerary looks like this:

Day 1 - Madrid
Day 2 - Seville
Day 3 - Seville
Day 4 - Seville
Day 5 - Seville
Day 6 - Granda
Day 7 - Granada
Day 8 - Granada
Day 9 - Madrid

We wont really see much of Madrid except for a few hours on each day, but I think we're okay with that. We dont want to hop hotels too often, and I feel like with the majority of time in SEville and Granada, we could always do a day trip in each city if we get bored

We’ve been to several cities in Spain in 2017 and enjoyed our trip even more than we expected. For the number of days you have available, and since you’re flying into and out of Madrid, I’d keep the itinerary you already had at our post. Córdoba, Ronda, Toledo, etc. were nice, but you would run short of time at the other locations.

By the way, we took a City Bike tour of Sevilla, and we would recommend it for a fun way to learn about the city’s history and see the neighborhood sites.

We stayed at both the Madrid royal palace neighborhood and the Retiro Park neighborhood for hotels in Madrid. Our favorite was in the royal palace neighborhood.

The Alsa bus between Granada and Seville is faster than the train and less expensive and was a pleasant ride. We bought tickets online while we were in Spain.