Rain is normal in Istanbul from November to March. The bad news is that Galata's cobblestones are properly slippery when wet. The good news is that the indoor places are all within a few minutes of each other — you can fill a day without getting soaked.
Morning. Walk down to Galip Dede Caddesi. The street is lined with instrument makers, all of them small shops, all of them indoors, and browsing is welcome. Watching someone tune a bağlama is a good way to spend a wet morning.
Towards midday, go to the Galata Mevlevi Lodge at the top of Tünel, set around a closed courtyard. It is a small museum — half an hour is enough — but the wood of the ceremonial hall and the carved headstones in the garden make that half hour stay with you.
Lunch in Karaköy. Five minutes downhill, ten on wet stone. The fish restaurants under the bridge are enclosed and warm.
Afternoon, go into Salt Galata, the old Ottoman Bank building. The library and reading room are open to anyone, there is a floor where you can have a coffee, and you can sit inside for hours. Entry is free.
Late afternoon, take the Tünel funicular back up. Two minutes, under cover, and it saves you the climb — reason enough on a wet day.
Two practical notes: do not wear flat-soled shoes, and a hooded coat beats an umbrella. The wind off the Bosphorus enjoys turning umbrellas inside out.