What is the right word here?
Traditionally, fathers usually give names to their babies after 3 days of their birth\arrival\delivery.
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Sign up to join this communityTraditionally, fathers usually give names to their babies 3 days after their birth.
Of those three, birth sounds most natural; babies don't arrive and delivery describes what the mother does (or the doctor/nurse), not the baby, who is being born, so an alternative would be
Traditionally, fathers usually give names to their babies 3 days after they're born.
Note that I've changed the word order for 'after' as well, the 'after' ... 'of' construction is only used when describing a prolonged event, e.g. "After 3 days of marching we reached the camp."
Also, I think you can drop the "usually", since it's already implied by "traditionally".
I would change to
Babies are traditionally named by their fathers when they are 3 days old
Although this can lead to ambiguity as to who is 3 days old in the sentence, it would be natural to assume the subject is the baby being named and not the father who is 3 days old.
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