NCC-1685 was the registry number of a Federation starship that was in service with Starfleet in the mid-23rd century. This unnamed starship underwent a layover at Starbase 11. The ship's number was seen on a chart indicating the repair status of several vessels. (TOS: "Court Martial")
The Eagle was actually mentioned in an early 1967 draft shooting script for TOS: "Journey to Babel",  and concurrently included in internal memos from D.C. Fontana and Robert Justman, suggesting lists of possible starship names to consider for use in The Original Series. These lists, published in The Making of Star Trek (pp. 164-165), were used to match Eagle with the registry NCC-1685 that appeared on the wall chart seen in "Court Martial" by Jein in his article.
A quarter of a century later however, Mike Okuda had a different number (NCC-956) assigned to Eagle in a graphic in Star Trek VI: The Undiscovered Country, leaving the "NCC-1685" registry orphaned.