What is a word that describes the start or opening of a conversation?
Solution 1:
A conversation opener.
Solution 2:
Make a conversation strike up.
strike up: to initiate, begin: to strike up a conversation
You can also kick off a conversation.
kick off: to start out, begin
Alternately, consider conversation starter.
Solution 3:
I like how Dr. John Gottman calls these a "bid for connection". Or just a bid. Bids can be verbal like, "You would not believe the day I had". Or they can be nonverbal, like, sighing, or sitting next to the person, or sending them a picture or link they might like.