The name is a mistake that stuck. These points are Greek, they are called lots, kleroi, the word for what falls to someone by allotment, and they are in the surviving Hellenistic material from the first centuries of the common era, several hundred years before there was any Arabic astrology at all. Ptolemy has Fortune. Valens, Paulus and Dorotheus have a great many more.
What happened is a transmission accident. The Greek corpus passed into Persian and then Arabic hands, was extended considerably there, and returned to Latin Europe in the twelfth and thirteenth centuries through translation from Arabic. Medieval European astrologers received the technique from Arabic books and named it after the books, calling the points partes and then, in English, parts. The label records where Europe got them, not where they came from.
The Arabic-language contribution was real but it was expansion rather than invention: the lists grew from a compact Hermetic set to a hundred and more, one for nearly every question that could be asked of a chart. Much of that expansion is mechanical, take the significator of a topic, take the significator of its opposite, project from the Ascendant, and the sheer number of lots is one of the standing objections to the technique. A chart with a hundred and fifty sensitive points in it has a point near everything.
The modern revival has largely gone back to lot as the term, partly for accuracy and partly to mark a return to the Greek sources rather than the medieval lists. Both names are in use. This site retains the familiar English label for findability and notes that it is historically wrong.