taru-adjective, adverb taking the 'to' particle
earnestly and patiently; repeatedly and persistently
Describes the manner of explaining, admonishing, or advising someone with earnest repetition and patience. Often used in literary or formal contexts with the と particle.
先生は生徒に諄々と注意を与えた。
The teacher patiently and repeatedly gave the student warnings.
彼は諄々と説得を続けた。
He continued to persuade earnestly and persistently.
繰り返し is a common noun/adverb meaning 'repeatedly', but lacks the nuance of patient earnestness that 諄々 carries.
The exact derivation is uncertain. The kanji 諄 carries the meaning of 'repetitious' or 'tedious', and the reduplication reinforces the sense of repeated, patient admonition.