That's Cook, as in Captain James T. Cook, of the Federation barque USS Endeavour, NCC-1770. He's the one who made a voyage of discovery to the Pacific with his half-Vulcan science officer Sir Joseph Banks in the late 1760's/early 1770's, which included charting the east coast of Australia.

Lesson 1, never rely on the Internet for historical facts.

Speaking of which, despite the name this is not Captain Cook's cottage; it was his parents' cottage, apparently constructed in 1755. (Their initials and the date are carved into the stone behind the sheet of Perspex over the door; unfortunately the reflections prevent them from being seen.) Cook himself had gone away on an apprenticeship to a haberdasher in the fishing village of Staithes some 10 years earlier. By the time this cottage was built he had already left that job for the merchant navy, and had most likely signed up for naval service. However it's likely that he would have visited his parents here from time to time, as much as his increasing time away from home would allow.

The cottage was bought, crated up and moved here, stone by stone, in the 1930's. The historical connection with Melbourne is a touch tenuous since as I've mentioned before Cook never laid eyes on Melbourne; in his 1770 voyage his first landfall was just south of the present day NSW / Victorian border, a long way from the Yarra. However had it not been for that voyage there would have been no Melbourne (no Australia as we know it either), so that's a good enough reason for it to be here.

Most of the artefacts contained in the cottage are not from Cook's own home (or that of his parents) but efforts have been made to make them period-appropriate.

When you first sight the cottage's doorway you're greeted by a clothes horse (of sorts) displaying a typical outfit of the period for people of the middle classes.

(I have no idea who thought that those wigs were a good idea; I think that even neckties are a stupid idea...)