In the Potterwatch broadcast heard by Harry, Ron and Hermione in Harry Potter and the Deathly Hallows, all of the contributors' codenames start with "R". Lee Jordan is River, Kingsley Shacklebolt is Royal, Lupin is Romulus and Fred is Rodent or Rapier.
I understand the meanings of each codename (e.g., River refers to both the River Lee and the Jordan River), but is there a canon-based reason for the R theme?
Theme naming is a thing, both in real life and popular culture. Maybe JKR was inspired by the US Secret Service: the code names they assign to the President and his family always start with the same letter. For example the Trump family got Mogul (Donald) Muse (Melanie) and Marvel (Ivanka). The Obamas (and the Reagans earlier on) got the "R" theme: Renegade (Barack), Renaissance (Michelle) Radiance (Malia) and Rosebud (Sasha).
In universe, it's completely in character for the twins, who love to play word games and puns even at deaths's door :
‘Saint-like,’ repeated George, opening his eyes and looking up at his brother. ‘You see … I’m holy. Holey, Fred, geddit?’ ~ Harry Potter and the Deathly Hallows - Fallen Warrior
and use alliterations (Weasleys' Wizard Wheezes).
Lupin is no different, what with he and his friends coming up with names like 'Marauder's Map'. As for the Potterwatch, Lupin again chose a (were)wolf themed nickname, no surprise here.
The twins or Lupin probably came up with the "R" themed cover names (perhaps they chose 'R' for Resistance?) and Kingsley probably indulged them, selecting a codeame based on his own surname (King - Royal).