Charles Ellicott Commentary 1 Kings 1:33

"And the king said unto them, Take with you the servants of your lord, and cause Solomon my son to ride upon mine own mule, and bring him down to Gihon:" — 1 Kings 1:33 (ASV)

Gihon (meaning “breaking forth”) is clearly a place in the valley under the walls of Jerusalem. It is mentioned as having a watercourse, or torrent, diverted by Heżekiah when he prepared the city for siege (2 Chronicles 32:30), and as forming one end of a new wall “up to the fish gate,” built by Manasseh. However, its exact location has been doubted: whether it is on the west of the city, near the present Jaffa gate, or (as seems more probable) on the south, at the end of the valley called the Tyropœon, which runs through the city.

The Targums read Siloam in this context; this reading agrees with the latter supposition (that Gihon is on the south), which is also supported by its proximity to Adonijah’s feast at En-rogel, as implied in the narrative.