Boza is a very interesting fermented drink that could be described as something like a vegan yoghurt. It's usually generally available at winter time, but at the Vefa boza shop you can buy it all year round, when it's not available anywhere else.
The "drink" itself is unusual in that you might not enjoy it very much the first time you taste it. I didn't like it the first time I tried it since I wasn't expecting the combination of sweet and acidic flavours, and unusual texture. But now I really enjoy it. The interesting thing is that it gives an amazing energy/well-being boost, especially if you try it after a long day walking around the city. Just see if you can notice it after trying some. It was traditionally only served in winter due to lack of refrigeration at the time, but I think it's almost better as a refreshing drink/food in the summer.
The shop is also a mini tourist destination in itself - it's just a small place but it looks like it hasn't changed that much since the 1876 date when it was established. It's quite unique really.
A glass of boza (with or without cinnamon) is just 3TL, and you can sit in the place and drink it, or you can buy one of the larger containers to take-away (but they will need to be kept in a cool place. You can refrigerate it, but it may become slightly lumpy). You can also buy some sort of spiced Ottoman-style "sherbet" drinks and ice-cream etc., but the boza is the main attraction here.
The opening hours are fantastic - until midnight every day of the week, so you can pop in at 11pm on a Sunday night for your boza hit before going home :)
Note: Vefa also frequently have a stall in the Hippodrome area (with the obelisks) near the Blue Mosque during Ramadan and also other times when there are food stalls set-up for exhibitions etc.
Cons: Limited selection apart from main item, Area very quiet at night