The Hanoli system was a star system.
In the mid-23rd century, a subspace rupture was reported in the system. Some time thereafter, a Vulcan science mission attempted to seal the rupture with a pulse wave torpedo, but instead caused the rupture to expand exponentially. The entire star system, along with the Vulcans, was wiped out after which the rupture reached its critical mass and imploded, destroying itself.
This system was only mentioned in dialogue.
Despite the event being explicitly said to have happened in the mid-23rd century, when talking about the device the Vulcans used Miles O'Brien said the technology had much improved in the last two hundred years, implying them to be limited to 22nd century technology. The Star Trek Encyclopedia (4th ed., vol. 1, p. 325) dated the event to the year 2169.
According to the script, the pronunciation for Hanoli was "hah-NOH-lee". . The script also specifies that Dax knew about the fate of the system out of personal knowledge, not through the research she was doing at the time.