I think it’s very sweet that you have a family tradition Out of your options I like [name_u]Rhys[/name_u], [name_m]Jaymeson[/name_m], [name_m]Waylen[/name_m], and [name_m]Gryffin[/name_m]!
Here are some other options that fit your criteria:
[name_m]Clyde[/name_m]
[name_u]Flynn[/name_u]
[name_m]Bryton[/name_m] (might be too close to [name_u]Bryce[/name_u]?)
[name_m]Hayes[/name_m]
[name_m]Haynes[/name_m]
[name_m]Sayer[/name_m] (might be my favorite along with [name_u]Rhys[/name_u])
[name_m]Axyl[/name_m]
[name_u]Baylor[/name_u]
[name_m]Dayne[/name_m]
[name_u]Dayton[/name_u]
[name_m]Ely[/name_m]
[name_m]Edyson[/name_m]
[name_m]Ewyn[/name_m]
[name_u]Hayden[/name_u]
[name_m]Jayce[/name_m]
[name_u]Jay[/name_u]
[name_m]Jaymes[/name_m]
[name_m]Kayne[/name_m]
[name_m]Layton[/name_m] (my other favorite)
[name_u]Layne[/name_u]
[name_u]Taylor[/name_u]
[name_m]Payce[/name_m]
[name_u]Payton[/name_u]
[name_m]Rayden[/name_m]
[name_u]Raylan[/name_u]
[name_u]Rey[/name_u]
[name_u]Saylor[/name_u]
I hope that give you plenty of options! My favorites for you are [name_u]Rhys[/name_u], [name_m]Sayer[/name_m], and [name_m]Layton[/name_m]. Good luck!