(at around 20 mins) John Connor knows his father's name only from the records his mother left him. It is not made entirely clear how Skynet could have obtained that information too and how it can identify Kyle Reese by his face (something which is not known even to John). However, it's entirely plausible, given Skynet's access to global information, that it could have obtained such information from records of events shown in previous Terminator films, such as Kyle Reese's arrest record and subsequent filmed interview in Terminator; this would have given Skynet a picture of his face, which could then be used in conjunction with facial recognition software to identify him. Additionally, during the many interviews between Dr. Silberman and Sarah Connor in the mental institution shown in Terminator 2 - Il giorno del giudizio, Sarah Connor very openly talks about Kyle Reese and his role in future and past events. Between Kyle Reese's arrest records and Sarah Connor's medical records, Skynet would have all the information it needed.
Plot hole - Skynet should have never known about Kyle Reese. If it knew that Kyle was to be sent back in time to father John Conner, it could have prevented it by not creating the time travel device and not sending the Terminator. Kyle would not be able to travel to the past and John would not have been born.

Film

Terminator Salvation (2009)

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