You don't need much, but what you do need is specific. Here's the short list:
A headscarf (women) — the mosque provides them at the entrance, but they're thin, communal, and not always clean. Bringing your own is more comfortable and hygienic. Any scarf large enough to cover your hair and drape over your shoulders will do. You can buy lightweight cotton scarves for a few lira from vendors all over Sultanahmet.
Clothing that covers your shoulders and knees (everyone) — this is non-negotiable. No shorts above the knee, no sleeveless tops, no low-cut necklines. If you're visiting in summer and don't want to spend the whole day in long trousers, bring a light wrap or sarong in your bag. More detail on the dress code page.
Socks — you'll remove your shoes before entering, and you'll be walking on carpet in your socks (or bare feet). In summer the carpet can be warm; in winter it can be cold. Clean, comfortable socks are worth thinking about.
A plastic bag — the mosque provides bags for your shoes, but having your own makes it easier. A simple supermarket bag works fine. You carry your shoes with you inside the mosque.
A water bottle — the security queue can take 30 to 60 minutes in peak season with no shade. Staying hydrated matters, especially in July and August. You'll need to finish or cap your water before entering — no food or drink inside.
Your phone or camera — photography is allowed inside the Blue Mosque. No flash, no tripods, no selfie sticks, but phone and camera photography is absolutely fine. Some of the best interior shots in Istanbul come from inside this building.