For a lot of ideas, check Tom Brosnahan's excellent Turkey Travel Planner website, lots of free and good information. (Tom wrote the original Lonely Planet guides to Turkey - 1st - 7th edition, as well as Frommer's, served there as a member of the US Peace Corps, etc.)

We eschew the tours up the Bosphorus and laze up on the Şehir Hatları traditional big ferryboats and maybe go to Anadolu Kavağı, northernmost town on the Asian side, with a ruined castle with lovely Bosphorus and Black Sea views, lunch at a seafood restaurant, or maybe visit Sariyer, lunch and visit the Sadberk Hanım Museum just south (Sadberk Hanim Museum website).

It's more costly than in the early days when we did it, but we still enjoy, meet nice Turkish people (some families will want to share tea and simit bread. Mmm!)

Of course, there's tons to do in the old quarter, ferry to Üsküdar for lunch, etc. (This slightly naughty song about a businesswoman taking her male secretary to Üsküdar makes good listening whilst you think of taking the ferry across... Though American singer Eartha Kitt sang it with an interesting approach and here. LOL!)