Frankfurt 3 nights in October

What plans do you have for Day 4? Those might impact your Day 3 day trip decision.

Day 4 we fly to Cairo

Hi, just checking what days of the week you're in Frankfurt, as most museums are closed on Monday. And what time of year? On Day 3 you could easily get to Mainz, just outside Frankfurt for the day, or 1/2 a day. The highlights are the Greek Isis ruins underneath a department store, the Gutenberg museum (closed Mondays), and amazing Chagall stained glass windows (open on Mondays) . It's all walkable from the Mainz Römisches Theater station, see details in my trip report - https://community.ricksteves.com/travel-forum/trip-reports/6-nights-in-germany-mainz-heidelberg-heidelberg-was-gorgeous

You could do some old-world wine-town hopping on the Bergstrasse Route, which is served by direct train from Frankfurt's main station. It's just 39 minutes by regional train to Heppenheim, probably the most popular town on this route:

https://diebergstrasse.de/storage/2025/04/Holiday-Route-2024.pdf

https://ourworldforyou.com/a-few-hours-in-heppenheim-germany/

We arrive Monday October 13th and leave Thursday the 16th. Thanks for checking.

For meals, head over to Sachsenhausen to one of the traditional Applewine Pubs. Try Lorsbacher Thal, Ätschel, Dauth-Schneider, or Apfelwein Dax.
Other museums to visit, Judengasse Museum, Film Museum, Historic Museum, Jewish Museum, Städel.
The Frankfurt Palmengarten is beautiful this time of year. I do believe the Romantic Museum is open on Mondays as is the Goethe Haus.
Your stay is going to coincide with the beginning of the Frankfurt Book Fair, so your hotel prices may increase a lot.

For day trips, I would consider going to Rüdesheim, and then a cruise to St. Goar/St. Goarhausen. Return with the train. Get off in Rüdesheim again, ride the gondola over the vineyards, do some wine tasting in Rhein Wein Welt.
Alternative is to take the train to Eltville, then the bus up to Eberbach Monastery. Eltville is a beautiful town on the Rhein, with a lovely promenade along the river, often with wine stands.
If visiting a medieval town is more interesting, take the train to Büdingen. With massive stone towers and walls, and 100s of half-timbered houses, it is a photographers treat. Book a private tour there to take you inside the walls and up on the towers.
Hessen Park Open Air Museum and the Saalburg Roman Fort are good day trips. Train to Bad Homburg, then take bus #5 to go to either of these or both.

You've recieved some very good advice so far. Let me add a couple things.

If you like art don't pass the Städel Museum in Frankfurt. It's very good. The Hessisches Landesmuseum in Darmstadt has an excellent geology and paleontology exhibit. Bad Homburg is a very good day trip destination, as is Wiesbaden, both of which are very easy to get to by train.

Mid-October is when a lot of the river tour boats stop running, so if that's on your agenda plan carefully.

Plan for rain, October is when we start to have a lot of it.