I think they have been misinformed.
Irish Names by
Donnchadh O Corrain &
Fidelma Maguire has Eltin as a modern form of Eiltine, which is an Irish Gaelic name that meant either "a hind" or "a lively, nimble person". But the only derivation I can find for Elkin is a medieval English diminutive of
Elie,
Elijah, or
Ellen.