(for characters)
The question is in the title!!! Also surname ideas and middle name ideas would be appreciated.
[name_m]Beckett[/name_m] does sound a little more modern to me than [name_f]Persephone[/name_f] and doesn’t have a connection to mythology (that I know of).
But I still think they would work together because not all sibling names have connections anyway.
It might be cute to use a mythology name as a middle name for [name_m]Beckett[/name_m] to connect the names 
[name_m]Beckett[/name_m] [name_u]Artemis[/name_u]
[name_m]Beckett[/name_m] [name_m]Jove[/name_m]
[name_m]Beckett[/name_m] [name_m]Apollo[/name_m]
[name_m]Beckett[/name_m] [name_m]Icarus[/name_m]
[name_m]Beckett[/name_m] [name_u]Demeter[/name_u]
Other ideas
[name_m]Beckett[/name_m] [name_u]Layne[/name_u]
[name_m]Beckett[/name_m] [name_u]Jude[/name_u]
[name_m]Beckett[/name_m] [name_m]Crane[/name_m]
[name_m]Beckett[/name_m] [name_f]Marine[/name_f]
[name_f]Persephone[/name_f] [name_f]Rose[/name_f]
[name_f]Persephone[/name_f] [name_u]Wren[/name_u]
[name_f]Persephone[/name_f] [name_u]North[/name_u]
[name_f]Persephone[/name_f] [name_u]Bay[/name_u]
[name_f]Persephone[/name_f] [name_f]Hope[/name_f]
[name_f]Persephone[/name_f] [name_f]Grace[/name_f]
[name_f]Persephone[/name_f] [name_u]Sage[/name_u]
[name_f]Persephone[/name_f] [name_f]Dawn[/name_f]
Surnames
[name_m]Craine[/name_m]
[name_m]Dunn[/name_m]
Fields
[name_m]Grover[/name_m]
[name_m]Hughes[/name_m]
[name_m]Lovett[/name_m]
[name_u]Marlowe[/name_u]
[name_m]Hart[/name_m]
[name_m]Wells[/name_m]
[name_u]Tailor[/name_u]
[name_m]Olsen[/name_m]
[name_u]Ray[/name_u]
Myers
[name_m]Beckett[/name_m] and [name_f]Persephone[/name_f] dont necessarily go together, but you could still make it work. Maybe each parent named a child their favorite, or there is a big age difference so the naming style changed.
I also like the idea @EmaleeAnna had, of using a mythological middle for [name_m]Beckett[/name_m]!
Also, while [name_m]Beckett[/name_m] and [name_f]Persephone[/name_f] dont match perfectly, I do love [name_u]Beck[/name_u] & [name_f]Effie[/name_f] or [name_u]Beck[/name_u] & [name_u]Percy[/name_u]!
some suggestions:
[name_m]Beckett[/name_m] [name_m]Aries[/name_m]
[name_m]Beckett[/name_m] [name_m]Achilles[/name_m]
[name_m]Beckett[/name_m] [name_m]Endymion[/name_m]
[name_m]Beckett[/name_m] [name_u]Neptune[/name_u]
[name_m]Beckett[/name_m] [name_m]Atlas[/name_m]
[name_m]Beckett[/name_m] [name_m]Leander[/name_m]
[name_m]Beckett[/name_m] [name_m]Jupiter[/name_m]
I think with [name_m]Beckett[/name_m] as a brother, maybe a more current name would work as a middle for [name_f]Persephone[/name_f]!
[name_f]Persephone[/name_f] [name_f]Violet[/name_f]
[name_f]Persephone[/name_f] [name_u]Brook[/name_u]
[name_f]Persephone[/name_f] [name_f]Ella[/name_f]
[name_f]Persephone[/name_f] [name_f]Claire[/name_f]
[name_f]Persephone[/name_f] [name_u]Harper[/name_u]
[name_f]Persephone[/name_f] [name_u]Lark[/name_u]
[name_f]Persephone[/name_f] [name_f]Margot[/name_f]
[name_f]Persephone[/name_f] [name_f]Olive[/name_f]
[name_f]Persephone[/name_f] [name_u]Willow[/name_u]
I think any short last name would go well with [name_m]Beckett[/name_m] & [name_f]Persephone[/name_f]!!
I actually like them together quite a bit. They both sound posh and preppy, and if [name_f]Persephone[/name_f] has a cool modern nickname, it would match perfectly with [name_m]Beckett[/name_m].
Some combos
[name_f]Persephone[/name_f] [name_f]Lila[/name_f]
[name_f]Persephone[/name_f] [name_u]June[/name_u]
[name_f]Persephone[/name_f] [name_u]Bay[/name_u]
[name_f]Persephone[/name_f] [name_u]James[/name_u]
[name_f]Persephone[/name_f] [name_u]Clare[/name_u]
[name_f]Persephone[/name_f] [name_f]Cora[/name_f]
[name_f]Persephone[/name_f] [name_f]Laura[/name_f]
[name_f]Persephone[/name_f] [name_u]Quinn[/name_u]
[name_f]Persephone[/name_f] [name_f]Maeve[/name_f]
[name_m]Beckett[/name_m] [name_m]Henry[/name_m]
[name_m]Beckett[/name_m] [name_u]Riley[/name_u]
[name_m]Beckett[/name_m] [name_m]Ethan[/name_m]
[name_m]Beckett[/name_m] [name_u]Elias[/name_u]
[name_m]Beckett[/name_m] [name_u]Ryan[/name_u]
[name_m]Beckett[/name_m] [name_u]Wolf[/name_u]
Surname ideas
[name_m]Stone[/name_m]
[name_u]Steele[/name_u]
Grave
[name_m]Smoke[/name_m]
Waterhouse
Townsend
Towner
[name_m]Ames[/name_m]
[name_u]Wolf[/name_u]
[name_u]Wilder[/name_u]
[name_u]Day[/name_u]