According to Jonathan Swift's A Modest Proposal, Topinamboo is the name of a province in Brazil, used by the satirist (together with Lapland) to signify wilderness, remoteness and so on. There is also a tribe of the same name, also known as the Tupinambas.
Topinambour, meanwhile, is the French name for the Helianthus tuberosus, which our cold maids do Jerusalem artichoke call them. Here's what they look like:
And here's a recipe from St. Delia.
It's not clear why the French should have got it so wrong. The tubers were originally sent back to France by the explorer Samuel de Champlain (founder of Quebec City, and the man for whom Lake Champlain is named) in about 1605, from the area now known as Cape Cod, in I believe Massachusetts. He noted that the flavour of the unprepossessing root was similar to the artichoke, but the French name does not reflect that (nor does the alternative poire de terre, or earth-pear, by allusion to pomme de terre, or potato).
As that suggests, the Jerusalem artichoke comes from neither Brazil nor Jerusalem, but from North America. The name in English is a corruption of girasole, from girare, to turn, and sole, the Sun. In fact the tuber is a relative of the sunflower, called girasole in Italian, which as you probably know turns its face constantly towards the Sun.
Hence the other names for the Jerusalem artichoke: sunroot or sunchoke. The same derivation gives us heliotrope, which also means "turn towards the Sun". Helianthus means simply sun + flower.
The English name artichoke, incidentally, has links to the names chard (or cardoon), the French for thistle (chardon), cactus, and the fruit known as tuna grown in Sicily and elsewhere. The choke part of the word is thought to be related to the Italian for 'stump' (ciocco) or to the fact that the central core is hairy and inedible, but that seems unlikely.
*obviously the Jerusalem artichoke is not a fruit. But interesting tuber might have been slightly off-putting.