Note: We travelled in this region in October 1999.
We happened upon a few clustered around Tandogan Meydani near the university. This is a short walk from the train station. We didn't get online here but priced one - Argos - it was 1mTL/hr ($2).
Didn't seek it out but did hear that there's at least one in town.
There are more cybercafes here in a small area than we've seen anywhere. We counted 15 on two floors of one building! That'd be the Cumhuriyet Is Merkezi building on Cumhuriyet Caddesi just across from the main post office. They seem to all charge 500kTL/hr ($1). There are others outside of this building as well - scattered around town.
There are two cafes - Cafedoci@ charges 1mTL/hr ($2) and Nese Cafe (aka Goreme Internet Cafe charges 1.5mTL/hr ($3). In addition some tour agencies will sell you time on their one machine at 1m or 1.5m.
There are many in Sultanahmet charging from 600kTL to 800kTL. Cheapest we found was Otantik Internet Cafe, Alaykosku Caddesi (corner of Yerbatan, across from Sunshine Hotel). Some hostels provide access but tend to charge too much.
We spent most of our time staying across the Golden Horn ưn Beyoglu. There are a slew of cybercafes along and just off of Istiklal Caddesi. Cheaper than those in Sultanahmet, they range from 500kTL to 750kTL an hour. Mostly we used Virus Internet Cafe at Galatasary (Yeni Carsi Caddesi just south of Istiklal Caddesi). Interland on Balo Sokok is in a spacious loft but charges 750kTL.
There's apparently one near the Alaettin Bulvari circle but we didn't bother.
Internet Baruda on Ataturk Bulvari north of Hotel Harran is just four machines jammed side-by-side in a tiny space and the connection is a bit slow. But it's the only one we found. 500kTL/hr.