As jwodder has pointed out in a comment, in the UK (and presumably the wizarding community in the UK), it's not a legally binding requirement to give a child the same middle name as their parent. Your parent/s can choose whatever forenames they like, though you're stuck with the surname.
Ron appears to have been named after his relative Uncle Bilius (the one who died 24 hours after seeing a grim in his teacup), presumably his father's brother.
‘Talking about Muriel?’ enquired George, re-emerging from the marquee
with Fred. ‘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
Harry Potter and the Deathly Hallows
In the UK, it's (semi-)traditional to give children the names of elderly relatives that one hopes to
inherit from commemorate after their deaths.
You may wish to note that Arthur Weasley did give his name to his eldest son, William Arthur Weasley (AKA Bill) as did Harry James Potter's father, Albus Percival Dumbledore's father and Ginevra Molly Weasley's mother.