What does Drake mean?
English surname-turned-given-name from the Old Norse byname Draki or Old English Draca, both 'dragon', ultimately via Latin draco from Greek drakon 'dragon, serpent'. Flag: it coincides with the unrelated English word 'drake' meaning 'male duck', and the medieval byname's sense is sometimes read as 'dragon/standard-bearer (banner)', so the dragon reading is well-supported but not the only one.
What does Remington mean?
An English habitational surname from Rimington (Yorkshire/Lancashire), from Old English 'Riming' (boundary stream) + 'tun' (settlement, enclosure), i.e. 'settlement on the boundary stream'. Often loosely glossed 'place on a riverbank'; it also carries firearm-brand associations.