Could [name_f]Claire[/name_f] be an honor name for [name_u]Charles[/name_u]?
[name_f]Charlotte[/name_f] is the obvious feminine form, but it doesn’t fit and is a little too formal for us, plus it has gotten super popular. [name_f]Claire[/name_f] is a name we kept coming back to and I’m just not sure if it’s a fit as a take on [name_u]Charles[/name_u] or not?
It’s a bit of a stretch, but I could see it. The most important thing, of course, is that you and the person you are honoring understand it as an honor name.
Claire is not a variant of Charles. But the similar-sounding Caroline is – you might consider that one if you’re open to alternatives. 
[name_f]Claire[/name_f] isn’t a direct variation of [name_u]Charles[/name_u] so it is more of a stretch but to be honest, for me the fact it shares letters and syllables is enough for me 
I think it just depends if that’s enough for you
I think it works! I think the shared initial and similar sounds are enough to make it an honor name.
We like [name_f]Caroline[/name_f], but already have a niece with that name. [name_f]My[/name_f] DH and I are both the youngest in our families, so lots of family names have already been used on nieces or nephews, hence the progression to [name_f]Claire[/name_f]!
Would the [name_u]Clare[/name_u] spelling make it more relatable to [name_u]Charles[/name_u]?
I think it’s a bit of a stretch, but it works! [name_u]Charlie[/name_u] could also be a cute honor name 
I love [name_f]Claire[/name_f], if you wanted some other options what about [name_f]Charlize[/name_f]? Same meaning as [name_u]Charles[/name_u] and a little more obvious then [name_f]Claire[/name_f]. [name_f]Francesca[/name_f] is another name with the same meaning as [name_u]Charles[/name_u] so a subtle way of honouring him, it also has lots of nickname potential too.
I think if you like [name_f]Claire[/name_f] you could use it. You could do honor name by having matching initials instead of name variants as well.
Other [name_u]Charles[/name_u] variants just in case:
Carola/Carole/Carol/Carroll
Charla/Charlize/Charleston/Carlissa/Charisse
Chase/Charlene/Carla/ Carlotta/ Charlo/Carsten
Chip/Chesley
Is [name_f]Catherine[/name_f] a better fit to [name_u]Charles[/name_u]?
Honestly I don’t get the connection between [name_f]Catherine[/name_f] and [name_u]Charles[/name_u] at all. I think [name_f]Claire[/name_f] is less of a stretch than [name_f]Catherine[/name_f] (though I prefer [name_u]Clare[/name_u] for honoring [name_u]Charles[/name_u] since it has all the same letters inside Charles).
Though you could honor [name_u]Charles[/name_u] with initials if you wanted to. Like if the [name_u]Charles[/name_u] you wanted to honor was [name_u]Charles[/name_u] [name_u]Edward[/name_u] you could do [name_f]Catherine[/name_f] [name_f]Elaine[/name_f]. [name_m]Just[/name_m] another idea if you wanted to use [name_f]Catherine[/name_f] but make it a slightly more obvious honor
Have you considered [name_f]Lottie[/name_f] or [name_f]Lettie[/name_f] as an alternative to [name_f]Charlotte[/name_f]?
Other names with the same meaning as [name_u]Charles[/name_u]: [name_f]Blythe[/name_f], [name_f]Caddie[/name_f], Frances/Francesca.
I think it’s a bit of a stretch to be honest. I think name beginning with ‘Char’ would have more of a connection. [name_f]Charlize[/name_f], [name_f]Charmaine[/name_f], [name_f]Charlene[/name_f], [name_u]Charlie[/name_u], etc. But if you wanted to go with [name_f]Claire[/name_f] I agree with the other comments that the [name_u]Clare[/name_u] spelling looks more like [name_u]Charles[/name_u].