I'd want to be on the European side since that's where the vast majority of the traditional tourist sights are, and it's somewhat time-consuming to cross the Bosphorus. That would eliminate (for me) Uskudar and Kadikoy from Cat Lady's list. On my recent trip to Istanbul I headed north of the Golden Horn on many occasions, and I noticed a lot of attractive areas up that way. The combination of historic architecture and gentrification was appealing from a tourist perspective.
However, I want to say that I would not do what you are planning. I'd split the time between one location south of the Golden Horn, not too far from Topkapi and Sultanahmet, and one to the north. That's what I did (roughly 10 days and 10 days, though for your purposes having more time to the north might be better), and I was glad I had done so. Being south of the Golden Horn would be more convenient for a lot of the traditional sightseeing you'll probably want to do. It will save you a lot of back-and-forth travel time. But it's a rather intense environment (especially west of Topkapi), so I wouldn't have wanted to stay there the whole time. During my ten days there I knocked off a lot of museums and mosques. After I relocated to the north, I focused more on visiting picturesque neighborhoods, in addition to going to museums not located in the Old City. There's so much to do in Istanbul that, aside from one ferry trip up the Bosphorus, I only got over to the Asian side one time. On that day I walked around the area southeast of the Uskudar ferry dock and (farther north) the very cute Kuzguncuk neighborhood.
Istanbul is extremely hilly; some streets turn into steps, not always with hand railings. Turkey has a four-season climate, with the average December experiencing 2 to 3 snowy days. I'd want to consider the terrain and pavement in selecting a place for a long stay. I'd hope to be able to walk from my lodging to the nearest Metro station and bus stops without dealing with troublesome cobblestones.
There's lots of public transportation available in Istanbul, but the pieces don't always fit together seamlessly, so you'll find CityMapper sometimes suggesting routings with 3 or 4 legs. It may help you to know that CityMapper and Google, when offering routing suggestions for pedestrians, seem to ignore the funiculars. Learn where those are so you can employ them when suitable. Most of the time you pay for each transit leg separately. Tickets are cheap (generally TL 15 at the time of my visit except for some ferries), but it means keeping more money on your transit card (IstanbulKart) than you might anticipate needing. I found the vending machines for reloading cards sometimes tricky to use, so I was happy to add a good bit of value when I found one that worked. I had better luck using cash (no large notes) rather than a credit card; only some machines take cards.
You shouldn't drink or brush your teeth with tap water in Istanbul, so be sure there's at least a little market close to your lodging. It's a drag to carry large bottles of water long distances, day after day. At least they are very cheap.
