The best districts to stay in Istanbul are Taksim-Beyoglu, Sultanahmet, Princess Islands and somewhere along the Bosphorus.
Taksim-Beyoglu
Pros: Culture and entertainment hub of Istanbul filled up with beautiful 19th century buildings. Many bars, restaurants, clubs, art galleries, museums, shops and many people. Contemporary and authentic. Full of energy. Transportation hub of Istanbul, so you can reach anywhere easily. Sultanahmet is only 15-25 minutes tram ride away. Many accommodation options.
Cons: None
Sultanahmet
Pros: This districts host the most important and popular attractions in Istanbul such as Blue Mosque, Hagia Sophia and Basilica Cisterns. Many accommodation options.
Cons: Low energy, not much to do in the night, many tourists. Too touristic shops and restaurants (high prices, hassling staff).
Princess Islands
Pros: Idyllic. No motor traffic is allowed. Green and quiet.
Cons: Very limited accommodation options. Reaching important parts of Istanbul is time consuming.
Places along Bosphorus
Pros: Being close to Bosphorus is a privilege.
Cons: Very expensive. Depending on where you are on the Bosphorus you might be far away from main attractions of Istanbul. Very limited number of places to stay and they are very expensive.