Some of these don’t meet every criteria, but they sure picked a difficult letter to match.
Hero Hawk –my top pick for them, despite not being two syllables
HeathHyacinthHalcyon HarmonyHeronHunterHaleHalo
Handsome
Handy
Harbor
Harlequin
Hart
Holland –probable
Hawkeye
Hawthorn –possible
Hazelnut
Hedge
Herald
Hibiscus
Holiday
Hollywood
Honest
Horizon
Humble
And I bet the kid is named
Mason or
Noah in the end, lol.