The OP was asking about the meaning in song choruses. In that case I can trace it back to the early 18th century in England when the word was then 'RIGDUM' and probably related to 'rig', a prank, a bit of fun or a trick.

The series of songs based on the tune and tricky chorus that survive in some variants of the 'Froggy Went a Courting' song and the minstrel song 'Kemo Kimo'. Which of these two used the tune and chorus first is hard to say. Froggy of course in different forms goes back into the mists of time.

The Camp Garland contains the song 'The Oxford Milkmaid's Entertainment at Camp at Burlington-Green. To the tune of 'Gameorum'.
This tune title suggests that the form, tune and chorus were already in existence when this ballad was made in the early 18thc.

The first verse runs

A Milk maid going out of late,
With a Rigdum for a little Game,
To see the Camp it was her Fate,
With a Rigdum for a little Game,
Gameorum wildum gorum, game wildum game,
Can you net a Flummerum, a Rigdum Bolleram,
Rigdum for a little Game.

At least some of these are real words and earlier usages of the chorus may have made more sense. If pushed I'd say it was probably camp followers' cant. The addition of 'um' onto the end of real words is a common form of 'in' lingo practised by some people, similar to the adding of 'y' and such onto the end of pet names and terms of endearment.

There are 26 stanzas of bawdy fun and frolicks with the officers and men.