What is the origin of the word "geroff"?

Solution 1:

but recently I've found sources claiming it is somewhat originating from "get off".

This is true, although it is not "somewhat", it is "entirely". https://en.wiktionary.org/wiki/gerroff#English and https://www.hp-lexicon.org/thing/geroff/

It is can be written as "Gerroff" as the pronunciation may include a slight "r" sound in the first part and a more noticeable one in the second part - "Ger'roff" - /gɛ(r)rof/

Gerroff is a phonetic spelling of "get off" when spoken quickly and/or in an annoyed manner.

Is there a linguistic rule that describes such a transformation?

"phonetic spelling".

Solution 2:

In some accents of British English, a word-final /t/ sound after a short vowel can be pronounced as something that sounds like an /r/ sound when the following word starts with a vowel. This is discussed on John Wells's phonetic blog; I quoted the relevant portions in a previous answer.

"Geroff" is a respelling of the pronunciation of "get off" in such an accent. (I don't know whether the specific form "geroff" might have spread to other speakers who don't have this as a usual feature of their accent.)

The British English "t-to-r" sound change is somewhat similar to the American English phenomenon of "flapped" or "tapped" /t/, but also different in several ways. Wells suggests that "t-to-r" developed by way of /t/ leniting in this position to a voiced flap [ɾ], which was then perceived as /r/, causing it to to be subsequently replaced for some speakers with the now more common [ɹ] allophone of /r/ (just as in words like "very", the [ɾ] allophone of /r/ seems to have become less frequent compared to the [ɹ] allophone).

American English speakers perceive a "flapped/tapped" /t/ as /d/, never (to my knowledge) as /r/. And American English speakers seem to use flap /t/ in more positions: inside of words, and after long vowels/diphthongs. So an American would be more likely to respell "Get off" as "Ged(d)off". Also, from the comments section of the John Wells blog post, it sounds like some British English speakers might have both a t-to-r rule and a separate flapping rule.

Solution 3:

From what I know and grew up with, "Gerroff" always meant get off and tended to be from the East Midlands dialect. And we used it sober and drunk, because the further north in England you go the dialect is presumed to get lazier according to some people! For example, in Yorkshire "go to the pub" ends up being shortened to "t' pub".