Septimus from L. septum "seven," probably a reference to Rowling associating the Weasley family with the number seven:
The backstory with Ginny was, she was the first girl to arrive in the Weasley family in generations, but there's that old tradition of the seventh daughter of a seventh daughter and a seventh son of a seventh son, so that's why she's the seventh, because she is a gifted witch. I think you get hints of that, because she does some pretty impressive stuff here and there, and you'll see that again (TLC).
Weasley="In Britain and Ireland the weasel has a bad reputation as an unfortunate, even malevolent, animal. However, since childhood I have had a great fondness for the Family Mustelidae; not so much malignant as maligned, in my opinion." (JKR) This genus includes weasels, stoats, polecats, ferrets, mink, wolverines, badgers, skunks, otters, and others.
Cedrella Black's marriage to Septimus Weasley caused her to be "blasted" from the Black Family Tree (BFT). While there is no direct statement in canon verifying that Septimus and Cedrella are Arthur's parents, the evidence is very strong. According to the Lexicon's essay "Arthur Weasley’s Relationship to Sirius Black: First Forays into the Black Family Tree":
Since Arthur’s parents’ names are unknown, we cannot directly link him to anyone on the Black family tree. There is one Weasley (not) appearing on the tree: Septimus Weasley, who married Cedrella Black, causing her to be blasted off the tree. Although it is possible that Arthuris related to Sirius through Arthur’s mother’s side (whose maiden name is unknown), it is reasonable to assume that Arthur is most directly related to Sirius through Septimus Weasley.