(written from a Production point of view)
This ship was identified by a Constitution-class icon in two display panels created for Star Trek IV: The Voyage Home. Two unnamed Constitution-class ships are seen docked at the Spacedock at the end of the film. It was assumed this ship was one of them. The panels were not clearly visible on screen in any scene of the final release, if present at all. These two panels show NX-2000 and the NCC-1707, with their matching class silhouettes, departure routes from the Spacedock. One of the panels abbreviated the registries to NX-200 and NC-177. These graphics were most likely created to correspond with dialogue from a deleted scene found in the shooting script for the two dock controllers: "Excelsior and Intrepid are cleared to depart. Open spacedock doors" followed by the failure of both ships to power up and the doors to open, due to influence by the Whale Probe. (Scene 27)
According to the in 1985 FASA published Federation Ship Recognition Manual (second edition, p. 11) the refit-USS Intrepid had the registry NCC-1707. At that point in time the FASA print materials were still considered official reference works and they were known to have occasionally been used as production use sources by among others Michael Okuda, specifically for his okudagrams. Okuda's very first job on live-action Star Trek actually concerned The Voyage Home as "Computer Animation and Tactical Displays" designer (a function later rechristened as "scenic artist", and which Okuda continued to perform for the franchise the next two decades), and who could have very well been the designer of the two unused panels in the first place.
Due to the fact that the designation was undiscernible onscreen, the registry NCC-1707 has remained non-canon.
Star Fleet Technical Manual stated the registry NCC-1707 belonged to USS Hood. This would contradict the registry NCC-1703 given for that ship in the 2008 remastered version of "The Ultimate Computer". While once considered "official", the Manual, like the FASA print material for that matter, has as of March 2002 been demoted to apocrypha status by the franchise. (Star Trek: The Magazine Volume 2, Issue 11, p. 71)
In Diane Duane's 1984 novel My Enemy, My Ally, USS Intrepid (NCC-1707) was mentioned as the original configuration replacement for USS Intrepid (NCC-1631), lost in TOS: "The Immunity Syndrome", meaning that FASA's USS Intrepid (NCC-1707) would be the refit-configuration of the same vessel, and therefore a very good candidate for the one mentioned in the deleted scene of The Voyage Home.