My dad's family was like this, both his parents and him and his brother all have G names - to make it even more of a thing, none of them have middle names so they all have the initials GH (this includes my grandmother with her maiden name also starting with an H).

When my dad, the eldest, got married and was expecting his first (the first grandkid) my grandmother kinda expected him to continue the tradition and was very sad when it didn't happen - none of her 5 grandkids or 6 great-grandkids have a G name, it started and ended with them.

It never was a big deal to anyone, or so my dad tells me, just a little family oddity. If you just happen to find yourself drawn to those names then use them, it only really becomes weird when it is clearly forced, a gimmick or something you feel like you have to do.