There are a lot of Weasley cousins:
The plan was to introduce Harry as “Cousin Barny” and trust to the great number of Weasley relatives to camouflage him.
"Another Weasley? You breed like gnomes!"
So there are a lot of Weasley cousins. Assuming all - or at least some of them - live in Britain, why don't any of them show up at Hogwarts?
Direct cousins, there is nothing I can find. Arthur had 2 brothers, and Molly had 2 brothers. Because Molly is the youngest, I can gather 2 possibilities from this. As Fabian and Gideon Prewett died from being in the first Order of the Phoenix, they might have not had any children or married. An alternative of this is that, as they were the older brothers, they had children before Molly did, and they had finished school as Mithrandir said in the comments.
On Arthurs side, not much is known about his 2 brothers, however if you believe Uncle Bilius (who saw the grim and died 24 hours later) is his brother, you could argue that he had no marriage, no kids and thus no cousins.
"Yeah, she’s just told me my ears are lopsided. Old bat. I wish old Uncle Bilius was still with us, though; he was a right laugh at weddings." "Wasn’t he the one who saw a Grim and died twenty-four hours later?" asked Hermione. "Well, yeah, he went a bit odd toward the end," conceded George. "But before he went loopy he was the life and soul of the party," said Fred. "He used to down an entire bottle of firewhisky, then run onto the dance floor, hoist up his robes, and start pulling bunches of flowers out of his -" "Yes, he sounds a real charmer," said Hermione, while Harry roared with laughter. "Never married, for some reason," said Ron. - DH
Furthermore, you could argue they did have cousins, mainly pure bloods like the Malfoys. This is due to pureblood families interbreeding, and as the Weasleys are listed as a pureblood family, you could say that they are related to the other purebloods at Hogwarts.
"The pure-blood families are all interrelated," said Sirius. "If you’re only going to let your sons and daughters marry pure-bloods your choice is very limited; there are hardly any of us left. Molly and I are cousins by marriage and Arthur’s something like my second cousin once removed. But there’s no point looking for them on here - if ever a family was a bunch of blood traitors it’s the Weasleys." - OoTP
Additionally, they might have just not gone to Hogwarts. Not until the 7th book where attendance of British children into Hogwarts was compulsory, but we don't know much about that as we followed Harry into hunting Horcruxes (emphasis mine).
"Attendance is now compulsory for every young witch and wizard," he replied. "That was announced yesterday. It’s a change, because it was never obligatory before. Of course, nearly every witch and wizard in Britain has been educated at Hogwarts, but their parents had the right to teach them at home or send them abroad if they preferred. - DH
