Breakfast in Istanbul is not a meal, it is a plan. Where you go depends on how much time you have.
The spread — serpme kahvaltı — fills the table with small plates: cheeses, olives, honey, clotted cream, jam, tomato, cucumber, sausage, eggs. A spread for two takes two hours and removes any need for lunch. It is for a morning with nothing after it; at weekends a table is hard to find after eleven.
Menemen arrives in one pan: tomato, pepper, egg. Ten minutes, cheap, and the right thing to eat before a day of walking. The test of a good one is that the tomato is properly cooked; if it is watery, it was rushed.
Just coffee. Karaköy and Galata are where third-wave coffee first took hold in Istanbul. Filter or espresso and a simit alongside is the most practical start if you are walking. Turkish coffee comes after breakfast, not instead of it.
Where. Downhill towards Karaköy is closer to the fish restaurants and the coffee houses; around the Galata Tower is touristed, but prices return to normal one street back. The small places around Serdar-ı Ekrem are both cheaper and better than the ones on the main run.
How much. A spread costs about what a mid-range lunch costs per person; menemen and tea are a third of that. Cards work everywhere for breakfast, but carry some cash for tips in the smaller places.
One tip: order tea by the glass rather than by the pot — a pot is not served for fewer than two and it goes cold.