I would eliminate:
[name]Henry[/name] [name]Rhys[/name] - sounds like the same thing twice and doesn’t flow as well as your others.
[name]Henry[/name] [name]Elliot[/name], [name]Henry[/name] [name]Elias[/name], and [name]Henry[/name] [name]Ernest[/name] - since [name]Henry[/name] ends with an “E” sound, I would avoid these as they run together.
[name]Henry[/name] [name]Joel[/name] and [name]Henry[/name] [name]Luke[/name] - the flow feels off here.
I like:
[name]Henry[/name] [name]Elliot[/name] [name]Kenyon[/name]
[name]Henry[/name] [name]Elias[/name] [name]Kenyon[/name]
[name]Henry[/name] [name]Pierce[/name] [name]Kenyon[/name]
[name]Henry[/name] [name]Rhys[/name] [name]Kenyon[/name] (I love both [name]Henry[/name] and [name]Rhys[/name] and they would have been in the love column but they are a bit of mouthful together.)
[name]Henry[/name] [name]George[/name] [name]Kenyon[/name]
I dislike:
[name]Henry[/name] [name]Joel[/name] [name]Kenyon[/name]
[name]Henry[/name] [name]Ernest[/name] [name]Kenyon[/name]
I disagree that [name]Henry[/name] [name]Elliot[/name] and [name]Henry[/name] [name]Ernest[/name] run together. [name]Henry[/name] ends in an EE, while [name]Elliot[/name] starts with an [name]Ell[/name] sound and [name]Ernest[/name] starts with an Urn sound. [name]Henry[/name] [name]Elias[/name] might run in together depending on how you pronounce [name]Elias[/name].
[name]Henry[/name] [name]Reese[/name] is my second favourite combo, so [name]Henry[/name] [name]Rhys[/name] gets my first vote!
Also I like [name]Luke[/name], [name]Elliot[/name] ([name]Eli[/name]/[name]Elliot[/name] [name]Henry[/name] was on my list once) and [name]Alexander[/name] as well.