The location and value on this stay is truly amazing. If you are looking to experience all of Istanbul and not just the sights of Sultanahmet then Besiktas is a wonderful place to stay. You are a student, younger center that has a wonderful energy (I never felt jet lagged with the energy outside even after a 14-hour flight). At the same time, the apartment is very quiet. From Besiktas you can easily get to the sights of Sultanahmet and Beyoglu, but you are also much closer to the real Istanbul life in Besiktas, Arnavutkoy, Bebek, Nisantasi.
I extended my stay twice and found Bora and Gulsah to be incredibly accommodating hosts who were also very friendly. We sat out and had a beer/cay multiple times and I really enjoyed meeting them.