Your right there is/was (iunno if it's still going) a Dark Horse character title called The Rook but, well, Rook (chess piece names in general) aren't super uncommon so I'm not surprised. However, that character's name is THE Rook, I would advocate just for Rook and lean more on a rook as a bird aesthetically as opposed to the easy chess route. I would also make his primary color green to make him more visually distinct and I want each bat character to have a distinct color (Nightwing blue, Red Hood, Signal yellow, Orphan black/gold, etc.).
Your never too old for Kid so long as your youthful, plus It's more of an editorial name, he'd mainly just be Conner (or occasionally Kon-El). Some of the names you suggested sound pretty cool but (and mind you I'm not super in-depth with Superboy) my interpretation of Conner is all about maturity and growing out of your flashy teeny-bopper phase into a responsible adult. So in my mind what better way to demonstrate that than to have him stand tall and just be who he is. His wife and child are from a different dimension and he doesn't have an earth civilian life due to being ejected from the earth for so long. I mean, what better way to distinguish him from Superman than to have him take Clarks open face mentality as Superman and evolve it to full transparency?