We are struggling with coming up with a boy's name we like for our next child, a boy, due in the spring. Our son is named after his grandfather's, Henry Palmer--and, we actually used his grandfather's middle names because their first names DID NOT work It's a classic name and we discovered after his birth and naming that it was becoming quite popular in our demographic, much to our annoyance. That said, we still love his name and it suits him perfectly. We want our next DS to have a name that's equally connected to both of our families. BUT, the thing is, the only names we have on the male side of our families that we liked we used for our older son. Thus our options are--using a male name w/o the familial significance (which I'd prefer not to do); using a male name (of a family member) that we don't like; or male names of family members we never really knew or felt connected to; or names that NO one else would like. We have three main criteria--we must have a "reason" for the name (eg family name; favorite disciple) and it must go with Henry and our last name (rhymes with Virus, starts with K sound).
That said, these are the family names we have to work with: Kermit, Gordon, Forrest, Bartholomew. So far, our list for consideration includes: Peter, Kermit, Bartholomew and Forrest. Peter would be our first choice by a long shot, it's the name of my partner's favorite disciple, but it also happens to be the name of a disliked, distant relative. We thought about going with the names of favorite theologians--but my favorites are Reinhold and Martin (Buber), neither of which really work for us. Oh, and if we did go with one of the odder choices I worry that our next child may be resentful that his brother got a perfectly normal, mainstream name (which is why we aren't just going with Kermit...).
I know you're flooded with questions but if you have the time (and we're interesting enough)...
I can see how you're stuck: you want a name that has family significance AND that you love---but there AREN'T any names that meet both preferences. You find the name of someone you love, but you don't love the name; so then you find a name you love, but it doesn't have the right kind of family significance. At such times I think we have to stop spinning, throw our hands up, and say, "What we want does not exist. And so, given that situation, what is our second choice?"
I think it's going to come down to which is more important to you: using a name you love, or using a name with positive family significance. This is a common choice to have to make: it's rare that the people important to us coincidentally have names that match our own personal tastes. This is why those families with a Grandma Ellie are all fighting to honor her, while Grandma Mildred's name goes completely uncontested---despite the fact that Grandma Mildred was awesome in every way and Grandma Ellie was an irritable old bat. It's one of the things that make honor names so honor-y: we use them for love and significance, not because we would have chosen the names anyway.
If I were you, I would go with a first name you love. The love becomes the reason for using it, and a very nice reason it is, too: "We gave you this name because we loved it so much." And perhaps you will find a name you love that has significance of a non-family variety, as with the idea of using disciple/theologian names. (I suggest Lewis, if you like C.S.; or George, if you like MacDonald.)
Then in the middle name position, I'd use the name of a beloved family member (if the child has one parent's surname, then I'd use a name from the other parent's family to give him the connection to both sides you're hoping for), accepting the dislike of the name as the cost of honoring someone important. Or, if you don't want to accept that cost, I'd use another name you love.
If the relative named Peter is a distant one, I think the favorite-disciple thing can trump it. If anyone in your family says, "Oh, after second-cousin-once-removed Peter?," you'd just say, "Oh! No, it's after Partner's Favorite Disciple." Peter is not such an uncommon name that there has to be an obvious connection, and Henry and Peter is a nice combination.
Edited to add: I just had one final thought: you mention looking at the names of male relatives, but are there any names of female relatives that would work (unisex names, names with feminine/masculine versions, surnames)?