From:The Vulgar Tongue - Buckish Slang and Pickpocket Eloquence by Francis Grose (London, 1785).

"GARRET ELECTION:- A ludicroud ceremony, practiced every new parliament: it consists of a mock election of two members to represent the borough of Garret (a few straggling cottages near Wandsworth, in Surry[sic]): the qualification of a voter is, having enjoyed a woman in the open air within that district: the candidates are commonly fellows of low humour, who dress themselves up in a ridiculous manner. As this brings a prodigious concourse of people to Wandsworth, the publicans of that place jointly contribute to the expence, which is sometimes considerable."

I don't know if it's worth mentioning, but "Tilbury" was a nickname for a sixpenny piece (being the ferry price from Gravesend to Tilbury Fort) - There may be a connection there.

The name "Garret" (or more usually "Garratt" still exists in the form of "Garrett Lane" the main road into Wandsworth from Tooting (like Wndsworth, now part of London, but once in Surrey).

Walrus.