I was a little surprised to find that the PIE root of do and δίδωμι is *deh₃-, not *do-. How did we get the "o" vowel sound from eh₃? I don't actually know how to pronounce h₃, but I'm assuming that *deh₃- rhymes with "meh"... correct me if I am wrong here.
This question popped up when I was looking up "anecdote" in a dictionary and reading its etymology. I saw that the word derives from ἐκ + δίδωμι (to give out, publish). The familiar root "do" is present. But when I looked up the origins of "do", I found *deh₃-.
(There you have it: an anecdote about anecdote.)
I might as well include a guess. My guess is this. *deh₃- became *di-deh₃-, but *di-do was considered a euphonic improvement somewhere down the line, hence δίδωμι. And, as Wiktionary suggests, the reduplication was lost in Latin, giving do.