My hubby and I want to choose a family name for both the first and middle name for our second baby. We don’t know the gender yet. Our DD is [name]Claire[/name] [name]Elizabeth[/name] (nn [name]Claire[/name] [name]Eliza[/name]), so we want a name that will work stylistically with her name. We don’t care about popularity of a name. Below are family names we are considering:
Girls:
[name]Catherine[/name], [name]Molly[/name], [name]Alice[/name], [name]Eleanor[/name], [name]Ruth[/name], [name]Mary[/name], [name]Anna[/name], [name]Margaret[/name], [name]Susan[/name], [name]Eva[/name], [name]Sarah[/name], [name]Victoria[/name], [name]Ellen[/name], [name]Agnes[/name], [name]Rose[/name], [name]Caroline[/name], [name]Jean[/name], [name]Emily[/name], [name]Julia[/name]
Boys:
[name]Oliver[/name], [name]John[/name], [name]William[/name], [name]Jacob[/name], [name]Scott[/name], [name]Henry[/name], [name]James[/name], [name]Peter[/name], [name]Thomas[/name], [name]Alexander[/name]
[name]Claire[/name], [name]Alice[/name] and [name]Rose[/name] all seem to go so well together. I chose [name]Victoria[/name] with a nod to [name]Claire[/name]'s mn.
[name]Claire[/name] [name]Elizabeth[/name] & [name]Alice[/name] [name]Victoria[/name]
[name]Claire[/name] [name]Elizabeth[/name] & [name]Rose[/name] [name]Eleanor[/name]
[name]Claire[/name] [name]Elizabeth[/name] & [name]Peter[/name] [name]Scott[/name] - I love [name]Claire[/name] and [name]Peter[/name] together! She would be [name]Claire[/name] [name]Eliza[/name] and he would be [name]Peter[/name] [name]Scott[/name]
[name]Claire[/name] [name]Elizabeth[/name] & [name]James[/name] [name]Alexander[/name]
[name]Claire[/name] [name]Elizabeth[/name] & [name]Oliver[/name] [name]James[/name]
[name]Catherine[/name] [name]Eleanor[/name] or [name]Catherine[/name] [name]Victoria[/name]. I like all the boys names
[name]Victoria[/name] [name]Catherine[/name] 
I love [name]Alice[/name] [name]Jean[/name]. It’s so sweet and simple.
[name]Thomas[/name] [name]Alexander[/name] sounds firm, confident and handsome.
With [name]Claire[/name] I like how it sounds with [name]Peter[/name], [name]Molly[/name], [name]Alice[/name], [name]Agnes[/name], and [name]Rose[/name].
[name]Peter[/name] [name]Joseph[/name], [name]Peter[/name] [name]Thomas[/name], [name]Peter[/name] [name]Henry[/name] or [name]Henry[/name] [name]Peter[/name]
[name]Molly[/name] [name]Victoria[/name], [name]Molly[/name] [name]Catherine[/name], [name]Molly[/name] [name]Jean[/name]
[name]Alice[/name] [name]Catherine[/name], [name]Alice[/name] [name]Juliette[/name], [name]Alice[/name] [name]Caroline[/name]
[name]Agnes[/name] [name]Ruth[/name], [name]Agnes[/name] [name]Victoria[/name]
[name]Rose[/name] [name]Evangeline[/name], [name]Rose[/name] [name]Genevieve[/name], [name]Rose[/name] [name]Catherine[/name], [name]Rose[/name] [name]Emilia[/name]
I like [name]Molly[/name], [name]Alice[/name], [name]Anna[/name], [name]Margaret[/name], [name]Eva[/name], [name]Victoria[/name], [name]Rose[/name], [name]Rosemary[/name], [name]Caroline[/name], [name]Emily[/name], and [name]Julia[/name] for girls.
I like [name]James[/name], [name]John[/name], [name]Scott[/name], [name]Thomas[/name], [name]William[/name], [name]Peter[/name] and [name]Jacob[/name].
I like your taste in names.
Combo might be;
[name]Eva[/name] [name]Margaret[/name],
[name]Julia[/name] [name]Rose[/name],
[name]Rosemary[/name] [name]Alice[/name]
[name]Jacob[/name] [name]Scott[/name]
[name]Thomas[/name] [name]William[/name]
[name]James[/name] [name]Alexander[/name]
Good [name]Luck[/name]!
With [name]Claire[/name], I like [name]Eva[/name] and [name]Henry[/name] the best as first names. [name]Alice[/name], [name]Rose[/name], [name]Thomas[/name], and [name]James[/name] are also quite dashing.
[name]Eva[/name] [name]Caroline[/name]
[name]Alice[/name] [name]Eleanor[/name]
[name]Rose[/name] [name]Victoria[/name]
[name]Henry[/name] [name]William[/name]
[name]Thomas[/name] [name]Alexander[/name]
[name]James[/name] [name]Oliver[/name]
These are all great suggestions, thank you! 
Oops meant to post this face
I like [name]Oliver[/name] [name]James[/name] or [name]Julia[/name] [name]Rose[/name]. I also really like [name]Margaret[/name] and [name]Eleanor[/name], but none of the other names on your list flow well with those as middle names [name]IMO[/name].