To have someone send a message or item via another person; to leave a message with someone (e.g., 'tell someone for me').
帰りが遅くなると伝えて、隣の佐藤さんに子供の面倒を見てくれるよう言付けた。
I told our neighbor Sato to look after the child, letting them know I'd be late.
書類は事務の田中さんに言付けておいたので、明日受け取ってください。
I left the documents with Tanaka in the office, so please collect them tomorrow.
彼に言付けておけば、忘れずに連絡してくれるだろう。
If I ask him to pass it on, he'll probably remember to contact you.
An archaic use meaning to use something as an excuse or pretext.
彼は天候を言付けて出席を断った（古い表現）。
He used the weather as an excuse to refuse attendance (archaic expression).