Argy-bargy is British slang with the meaning ‘noisy quarrelling’.
The word appears to come from an earlier form, ‘argle-bargle’, which originated in Scotland. The first part of the doublet is a modification of the word ‘argue’ and the second part is nonsense rhyming. Oxford lists the plural as argy-bargies.
This type of playful language is known as reduplication and English speakers love to coin them. Other examples include tittle-tattle, hoity-toity, mumbo-jumbo, hocus-pocus, super-duper etc.
Our love of rhyming jingles seems to be continuing, just think of hip-hop and bling-bling.