One of the very first things we learn about the Weasley family is that not all of them are wizards:
‘Are all your family wizards?’ asked Harry, who found Ron just as interesting as Ron found him.
‘Er – yes, I think so,’ said Ron. ‘I think Mum’s got a second cousin who’s an accountant, but we never talk about him.’
Harry Potter and the Philosopher’s Stone, chapter 6, “The Journey from Platfrom Nine and Three-Quarters”, p. 75 (Bloomsbury Paperback)
Over the course of the seven books, we of course learn a whole lot more about this family, but absolutely everything we learn about them (at least that I can think of) goes very much against the sentiment underlying this statement.
In particular, we learn that:
- The Weasleys are very family-oriented and put very high stock in the value of family—including family members they don’t particularly care for (like Aunt Muriel)
- Although mostly pure-blood, the Weasleys are completely devoid of feelings of supremacy based on their blood status
- At least one Weasley—Arthur—is in fact utterly fascinated and obsessed with all things non-magical, though the rest of his family (quite rightly) think he’s a bit barmy on that account (I mean, the man collects spark plugs…)
On the old ‘extra stuff’ section of JK Rowling’s website, there was a bit about Mafalda (Prewett?) who was, in an earlier draft of Goblet of Fire, meant to be this accountant1 cousin’s daughter (now gone from the website, but still findable on the Wayback Machine; link taken from this answer). That section mentions in a somewhat offhand way that,
This stockbroker had been very rude to Mr. and Mrs. Weasley in the past, but now he and his (Muggle) wife had inconveniently produced a witch, they came back to the Weasleys asking for their help in introducing her to wizarding society before she starts at Hogwarts.
As we all know, this plot thread never came to be, so we never really hear more about the accountant cousin, and his daughter doesn’t exist at all. But even if it had not been binned, it wouldn’t quite have explained Ron’s statement. Molly’s aunt Muriel had also been very rude to the Weasleys in the past, and most likely continued to be so until her dying day, but she is always present as a part of the Weasley family. She is, in fact, talked about rather a lot perhaps exactly because she’s so unpopular, particularly with the children.
Of course, a second cousin is not as close a relation as an aunt—but that’s no reason to actively “never talk about him” the way Ron implies is the case.2 Ron’s statement rather implies that the cousin’s job as an accountant is the reason he’s not talked about. Accountant cousin’s Wiki article interprets this as fact and states that his profession embarrassed the entire Weasley family.
But considering the Weasley family’s sense of blood-is-thicker-than-water and lack of blood-supremacy, combined with Arthur’s obsession with everything Muggle, why would being a Squib/accountant be a reason not to be talked about in the Weasley family?
1 Or stockbroker cousin. Doesn’t seem like Rowling ever really made up her mind about which he was, or perhaps 11-year-old Ron just didn’t know the difference. I’ll just call him an accountant here for simplicity.
2 If Ron had said something like, “Mum’s got a second cousin who’s an accountant, but we don’t see him much, what with him living in the Muggle world and all”, that would have been much more understandable to me, and wouldn’t have implied that accountant cousin is persona non grata in the Weasley household.