Should I say "To whomever gave me this gift: thank you" OR "to whoever gave me this gift: thank you"
'Whomever' is formally correct (as the object of the preposition 'to'), but many native speakers say 'whoever' informally.
Mr. Micawber:

I am NOT disputing your answer.

I was just wondering, however, whether the "correct" word is, in fact, "whoever."

It appears that the subject of "gave" is "whoever." The phrase "Whoever (the person who) gave me this gift" seems to be the object of the preposition.

In other words: I want to extend my thanks to whoever gave me this gift.

(You once correctly advised me to be VERY careful before posting an answer. I have given much thought to this post. If I am wrong, please do delete it. I do not want to be responsible for passing along bad advice. Thank you.)
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You are quite right, James—I overlooked the fact that it is the whole clause that is the object of the preposition, and that 'whoever' is the rightful subject of 'gave'. My answer was wrong.
To whoever gave me this gift, I would like to say, "Thank you."
What exactly is your point, Skype? Do you find anything wrong with the original poster's sentence? If so, could you explain it for us?
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"To whoever gave me this gift: thank you" is correct.

If anyone's interested, the grammar works like this:

"Whoever gave me this gift" is a fused relative construction. "Whoever" is called a 'fused' relative word because the antecedent and the relative word are fused together instead of being expressed separately as in simpler constructions.

"Whoever" is thus simultaneously head of the underlined noun phrase and subject of the relative clause; the meaning is comparable with "the person who gave me this gift". Since the non-fused "who" is subject of this simpler construction, it follows that "whoever, not *whomever, must likewise be the correct fused relative word.

Fused "whomever" would be strictly correct in, for example, "Invite whomever you like", although many people would just as correctly use "whoever".