“Going for an Eartha” is Cockney rhyming slang for “going for a shit”. The surname of Eartha Kitt rhymes with “shit”, creating this rhyming slang expression.
Cockney rhyming slang originated in the East End of London around the 1850’s. Rhyming slang phrases are created by taking a common expression that rhymes with the intended word and then using either the whole expression or just the first word to replace it. For example, the word “butcher’s hook” rhymes with “look”. “Having a look” therefore becomes “having a butcher’s” in rhyming slang.
It remains a matter of speculation whether rhyming slang came about by accident or whether it started as a secret language developed intentionally to allow market traders to collude or criminals to confuse the police.
“Give me five minutes, I need to go for a quick Eartha before we hit the road!”.