Whittaker means "white field,” which seems rather lackluster particularly if your other children have names with strong meanings. Benjamin’s meaning of “son of my right hand” (which designates high favor) is much more evocative.
Benjamin Whittaker has a nice flow.
Here are some other names with repeat letters and their meanings:
Cassian- uncertain origin perhaps from "protective cover"
Atticus- "shore"
Steffan- "crown"
Emmett- "whole" or "universal"
Phillip- "friend of horses"
Allen- "handsome"
William- "will protect"
Wyatt- "brave battle"
Elliot- "the Lord is my God"
Wallace- "foreigner"
Warren- "general; warden"
Matthew- "gift of God"
Brennan- "son of water"
Jesse- "gift"
Kenneth- "handsome"
Bennett- "blessed"