The story
"Wire" in the sense of telecommunications. The original waiver wire was "a network of telex communications where the status of all players is made known to all of the clubs".
Etymology from Wiktionary, CC BY-SA 4.0.
waiver wire
Today it means: a league-wide list to which an inexperienced player is added for a short period when cut from a team's roster; any team in the league may make a claim for the player during the period, at the end of which the lowest ranking claimant has right of first refusal, or the player may become a free agent if unclaimed Full definition →