The Gariman sector was a region of space.
En route to this sector, during the night shift, the USS Enterprise-D performed with its lateral sensor arrays three separate spectral analyses of the Alwanir Nebula on three distant frequencies. (TNG: "Rightful Heir")
According to the script, the pronunciation for Gariman was "GARY-man". In the script, Gariman was a star system. 
It's unclear if the Enterprise reached the sector before being rerouted to Boreth.