The Chora Museum/Mosque and the Basilica Cistern were both closed for renovation when I was in Istanbul in November.
Here are a few random things come to mind. They are sort of scattered, and so you will need to look at a map to see what appeals to you most.
Rustem Pasha Mosque -- a small mosque just with gorgeous Iznik tiles outside the Spice Market. Designed by Sinan.
Cistern of Theodosius -- smaller than the Basilica Cistern, but very nice. Recently, they started doing a sound and light show, which is not exactly my cup of tea, but the cistern itself is worth seeing.
*.Istanbul Archaeological Museum is my pick of the museums, but there are many other great museums.
- Visit a hamam. I have enjoyed Cagaloglu Hamam, but there are many others to choose from.
You are visiting the Asian side and the tour includes a Bosporus Cruise, and so you are probably pretty well covered for seeing the Bosporus already.