What is the origin of the phrase "eating on"?

I found this answer on a related post, which seems accurate and makes sense to me:

As a native speaker of a 'deep south' dialect, I believe I can provide a fairly authoritative answer.

Eat is inherently telic—unmodified it implicates (although it does not entail) complete consumption of its object. Consequently, a futurive construction such as I will eat or I'm going to eat is implicitly perfective.

On is added in my dialect to explicitly cancel those implications: the speaker claims that he will eat portions of the spaghetti at intervals throughout the coming week, without necessarily consuming the whole of it.

(However, "I'm going to eat ...", falls oddly on my ear; it suggests (again, in my dialect) that the speaker is announcing his firm intention of dining on the spaghetti. A more likely construction would be "I'm going to be eating on the leftover spaghetti all week.")

Eating on is not by any means a fixed idiom in my dialect. What we have here is, rather, a regionally widespread use of Verb on Object instead of bare Verb Object to provide an explicitly imperfective sense: for instance, we sand on a board for a while or consider on a topic without coming to any conclusion. This use of on is by no means unknown in Standard English (whatever the hell that is), where one may nibble on the hors-d'oeuvres or work on an assigned task; but the use is more widespread and the construction more productive in the South.

Originally posted by StonyB