Hotels for 1 night post cruise in Istanbul

You’ll be better off finding a hotel near the sights. My favorite is the Bosporus-view rooms with balconies at the Cihangir Hotel Bosporus on Cihangir Caddesi. You can drop your luggage off before check-in time and then begin seeing the attractions. You can also arrange for a taxi to the airport at any hour you need one through the taxi stand just down the street ( it was $20 when I was last there) or have the front desk arrange a taxi for you.
In the wee morning hours, the trip to IST airport takes about 35 minutes.

We are spending five nights in Istanbul at the same time as you, arriving 27th October and leaving 1st November.

We are booked into the Wings Hotel, in a Bosphorus view suite.

It isn't far from Galataport, walkable or there is public transport. It is surrounded by restaurants and walkable to the Galata Bridge and lots of other tourist areas.

We plan on private transfers to and from the airport. We were recommended to use www.istanbulairporttaxi.com.

https://www.thewingshotel.com/

Welcome pickups is what I used for airport transfer and it worked well.

Or should we stay at a hotel near all the attractions and risk taking
a taxi going to the airport at 3am for our flight at 6am. This is our
first time in Istanbul and we really want to see the iconic landmarks.

You should be at the airport 3 hours early if it's a flight back to the USA to ensure you do not miss your flight and have ample time to check bags and go through security and make the trek to your gate. Leaving at 3 a.m. from the historic area doesn't achieve that goal. Yes it is likely traffic won't be as bad as during the day, but it is still a solid 35-40 minute drive from the historic area. I am conservative and don't like missing a flight, so if it were me, I would find a place to store my bags for the day (it looks like that service is available at Galataport), do some sightseeing, and then take a shuttle to a hotel near the airport. Search the forum section for Turkey, there have been questions about airport area hotel recommendations.

BTW you'll need to confirm what is open on your single day there. Topkapi palace is closed on Tuesday, so you won't have that as an option.

Just a heads up that the Galataport terminal can be a bit confusing. It is located under a shopping mall. If you are going to grab a taxi, take from the ground transport area below the mall. If walking or taking the tram, then go to ground level and exit toward the street. It should be a few blocks walk to the closest tram station which can take you to most popular tourist destinations.

I stayed in what Rick Steves calls the Old City part of Istanbul last week. The hotel arranged a driver to the airport. At 4:30 am, with almost no traffic, the trip took about 55 minutes. Plan for the probability of a lot of walking inside the airport. I thought the early departure was worth it to have that final full day in Istanbul.

Thank you so much for all of your suggestions.

The ‘Easy Point Left luggage service’ at the Galataport might be a good option to look into.

Just wondering if anyone used that service before? Is it safe to leave our luggage there for 5-6 hours while we take the walking tour( Blue Mosque, Hagia Sophia and Basilica Sistern).
Thanks again!