[name]Hi[/name] there…I’d definitely stick with a traditional or antique name to work with [name]Jerry[/name] ([name]Jerold[/name]?) and [name]Nathaniel[/name].
BOY:
[name]Jackson[/name] (i like it, DH doesn’t): [name]Jackson[/name] isn’t unique (type the name into the box on the left to see its popularity), and it’s actually quite trendy now. I’d personally go with the less trendy (but very popular) [name]Jack[/name].
[name]Jourdain[/name] (we both like it, but not sure): For me, this yooneek spelling honestly isn’t classy at all, and [name]Jordan[/name] isn’t unique either. It’s actually pretty dated (1980s). (I’m just trying to help you out!) 
[name]Jesse[/name] (not fit with our last name): It doesn’t work with your last name, so we’ll skip it!
[name]Jeremiah[/name] (we have no idea what is the nn): With a son named [name]Jerry[/name], I’d stay away from [name]Jeremiah[/name], unless you want two Jerrys. [name]Even[/name] if you give him a nickname like [name]Remy[/name] (great idea, [name]Susan[/name]), both names begin with [name]Jer[/name].
[name]Jedediah[/name]: This one feels really [name]Beverly[/name] Hillbillies to me, but I do think it could work. Wait. It doesn’t flow well with your last name to me…
[name]Kai[/name]: I don’t think [name]Kai[/name] works with [name]Jerry[/name] and [name]Nathaniel[/name].
[name]Gray[/name]: Ditto.
[name]Clayton[/name]: From your list, I’m giving [name]Clayton[/name] my vote.
and we love [name]Caspian[/name] for MN, sounds strong and handsome: I think [name]Caspian[/name] is great!
GIRL:
[name]Alexandrina[/name]: [name]Alexandrina[/name] has that invented name feel to me, and doesn’t sound classy to me, which you’re understandably looking for in a name. All of the [name]Alex[/name] names are very popular/trendy now.
[name]Arianna[/name]: I think [name]Arianna[/name] is pretty, but it comes with pronunciation problems (Ahr-ee-on-uh, [name]Air[/name]-ee-on-uh, [name]Air[/name]-ee-ann-uh, etc.).
[name]Kassandra[/name] (nn [name]Kass[/name]): Spelllng beautiful C names ([name]Cassandra[/name]) with a K is honestly seriously trendy, and not classy/unique. I think [name]Cassandra[/name], spelled with its authentic spelling, is a lovely choice.
Boys J (I’d personally go with a different letter, so each child can have his/her own initial:
[name]Jude[/name] [name]Caspian[/name]
[name]Jonah[/name] [name]Caspian[/name] ([name]Jonah[/name] doesn’t really work with your last name, though)
[name]Jasper[/name] (doesn’t work with [name]Caspian[/name])
[name]James[/name] [name]Caspian[/name]
[name]Jefferson[/name] (not with [name]Caspian[/name])
[name]Julian[/name] (ditto)
[name]Non[/name]-J that work with [name]Jerry[/name] and [name]Nathaniel[/name] (I’d keep it traditional)…not all of them work with [name]Caspian[/name]:
[name]Atticus[/name]
[name]Augustin[/name]/[name]August[/name]
[name]Byron[/name]
[name]Calvin[/name]
[name]Conrad[/name]
[name]Charles[/name]
[name]Dashiell[/name] (pronounced dash-ill, nn [name]Dash[/name])
[name]Dexter[/name]
[name]Everett[/name]
[name]Elias[/name]
[name]Edward[/name]
[name]Gideon[/name]
[name]Leo[/name]
[name]Milo[/name]
[name]Maxwell[/name]
[name]Oliver[/name]
[name]Phineas[/name]/[name]Phinean[/name] (nn [name]Finn[/name])
[name]Silas[/name]
[name]Simon[/name]
[name]Samson[/name]
[name]Theodore[/name]
[name]Truman[/name]
[name]Wilson[/name]
[name]Xavier[/name]
Girl J:
[name]Jane[/name]
[name]Josephine[/name]
[name]Juliet[/name]
[name]Julia[/name]
[name]Jessamine[/name]
[name]Jocelyn[/name]
[name]Non[/name]-J:
[name]Amelia[/name] (popular, though)
[name]Beatrice[/name]/[name]Beatrix[/name]
[name]Cordelia[/name]/[name]Cora[/name]
[name]Celia[/name]
[name]Dahlia[/name]
[name]Daisy[/name]
[name]Eliza[/name]
[name]Evangeline[/name]
[name]Felicity[/name]
[name]Flora[/name]
[name]Genevieve[/name] (gives you the J sound, but a different initial)
[name]Georgia[/name]/[name]Georgiana[/name] (ditto)
[name]Helena[/name]
[name]Hazel[/name]
[name]Isla[/name] (eye-la)
[name]Iris[/name]
[name]Ivy[/name]
[name]Louisa[/name]
[name]Lydia[/name]
[name]Laurel[/name]
[name]Matilda[/name]
[name]Mirabel[/name]
[name]Mae[/name]
[name]Nora[/name]
[name]Phoebe[/name]
[name]Ruby[/name]
[name]Rose[/name]
[name]Rosalie[/name]
[name]Susannah[/name]
[name]Violet[/name]
[name]Viola[/name]
[name]Winifred[/name]
Good luck!
