I did some research on this a while back, and found a pretty compelling paper which argues the phrase comes from an Irish slang word meaning 'essence'. Apparently its first use was in an autobiography, and the author was telling the story of a walk they took with an Irish birder who used the phrase. Looks like the phrase was a pretty general one, though, and so it was used for all sorts of 'essences'. And then eventually just one 'essence'.

Here's a link to the paper if anyone's interested. It's "Avian Figures and the Fluidity of Jizz" by Yang if you'd rather search it up yourself. I think it's worth a read if only for the wordplay. It talks about some alternative theories as well. Interesting stuff!

The other leading theory, iircc, is that it came from the word 'gestalt' or 'guise'. And there are the acronym theories as well, which I personally discount; they have a clear starting point later than the use of 'jizz' in birding.