Alternate timeline
(covers information from an alternate timeline)

Lieutenant Barrett was a Starfleet crewmember aboard the USS Enterprise-D in an alternate timeline.

Barrett was paged to report to communications over the ship's intercom shortly after the ship encountered the USS Enterprise-C near a temporal rift in 2366. (TNG: "Yesterday's Enterprise")

This character was only mentioned in dialogue.
According to the Star Trek: The Next Generation Companion, the name Barrett was a homage to Majel Barrett.