DH is stuck on [name_f]Lily[/name_f] which I love, but I was trying to not have two nouns within her name so I suggested [name_f]Lyla[/name_f]. I started looking at other names and it turns out I am obsessed with the nouns for some reason…
[name_u]River[/name_u] [name_f]Lily[/name_f]
[name_u]River[/name_u] [name_f]Lyla[/name_f]
[name_u]River[/name_u] [name_u]Story[/name_u]
[name_u]River[/name_u] [name_f]Sonnet[/name_f]
[name_u]River[/name_u] [name_f]Clara[/name_f]
I like [name_u]River[/name_u] [name_f]Clara[/name_f] the best from your list . I think [name_f]Clara[/name_f] makes the modern [name_u]River[/name_u] more classic & feminine . [name_u]River[/name_u] [name_f]Lily[/name_f] seems like a flower to me . [name_u]River[/name_u] [name_f]Lyla[/name_f] is better .
Best of luck ! 
[name_u]River[/name_u] [name_f]Sonnet[/name_f] makes me sigh… oh, so lovely. I like the imagery embedded in that as well.
[name_u]River[/name_u] [name_f]Lily[/name_f] is a bit kitchy to me, but it technically works. [name_u]River[/name_u] [name_f]Lyra[/name_f] might be a decent alternative to that and [name_f]Lyla[/name_f]/[name_f]Lila[/name_f].
If you like the noun-names, here are a few pairings you might go for:
[name_u]River[/name_u] [name_f]Willow[/name_f]
[name_u]River[/name_u] [name_f]Celeste[/name_f]
[name_u]River[/name_u] [name_u]Vale[/name_u]
[name_u]River[/name_u] [name_f]Azalea[/name_f]
[name_u]River[/name_u] [name_f]Blossom[/name_f]
[name_u]River[/name_u] [name_f]Violet[/name_f]
[name_u]River[/name_u] [name_u]Gale[/name_u]
[name_u]River[/name_u] [name_f]Laurel[/name_f]
[name_u]River[/name_u] [name_f]Mavis[/name_f]
[name_u]River[/name_u] [name_f]Calla[/name_f]
I love [name_u]River[/name_u] [name_f]Willow[/name_f], too!
I prefer [name_u]River[/name_u] [name_f]Lyla[/name_f] over [name_f]Lily[/name_f]… BUT my favorite on your list is [name_u]River[/name_u] [name_f]Sonnet[/name_f]. It’s just so beautiful! I really wouldn’t look any further…
Other ideas anyways:
[name_u]River[/name_u] [name_f]Rose[/name_f]
[name_u]River[/name_u] [name_f]Pearl[/name_f]
[name_u]River[/name_u] [name_f]Elena[/name_f]
[name_u]River[/name_u] [name_f]Ivy[/name_f]
[name_u]River[/name_u] [name_f]Grace[/name_f]
[name_u]River[/name_u] [name_f]Gracelyn[/name_f]
[name_u]River[/name_u] [name_f]Sophia[/name_f]
[name_u]River[/name_u] [name_f]Genevieve[/name_f]
totally love the suggestion of [name_u]River[/name_u] [name_f]Rose[/name_f]! from your list I really like [name_u]River[/name_u] [name_f]Lyla[/name_f] and [name_u]River[/name_u] [name_f]Lily[/name_f]
some other suggestions:
[name_u]River[/name_u] [name_f]Kate[/name_f]
[name_u]River[/name_u] [name_f]Marie[/name_f]
[name_u]River[/name_u] [name_f]Amelia[/name_f]
[name_u]River[/name_u] [name_u]Valentine[/name_u]/[name_f]Valentina[/name_f]
[name_u]River[/name_u] [name_f]Seraphina[/name_f]
[name_u]River[/name_u] [name_f]Violet[/name_f]
I suggest, since you like [name_f]Lyla[/name_f], [name_u]River[/name_u] [name_f]Delilah[/name_f].
Or [name_u]River[/name_u] [name_f]Liliana[/name_f].
I dislike [name_u]River[/name_u] [name_f]Lily[/name_f] because it sounds like an actual type of plant. So, I would sway away from that. [name_u]River[/name_u] [name_u]June[/name_u], [name_u]River[/name_u] [name_f]Claire[/name_f], [name_u]River[/name_u] [name_f]Mae[/name_f], or [name_u]River[/name_u] [name_u]Ruby[/name_u].