But then, there exists a response...
Thanks, but no thanks!
Yes, you are right. 'Thanks but no thanks' is a response. It is an idiom. It is a phrase. It is an expression.
As such, the whole unit ('thanks but no thanks') expresses a certain meaning.
According to the free dictionary, the response is "A way of turning down something that is not very desirable."
Two examples are given:
Alice: How would you like to buy my old car?
Jane: Thanks, but no thanks.
John: What do you think about a trip over to see the Wilsons?
Sally: Thanks, but no thanks. We don't get along.
Other examples show this to be the meaning:
1 Thanks but no thanks: Pakistan PM shuns Indian limo for Nepal summit
Here, Pakistani Prime Minister Nawaz Sharif ... turns down an offer to borrow a bulletproof limousine lent by Pakistan's arch rival, India.
2 When the cooking was done, Margarita attempted to get me to eat the stuff. I said "Thanks but no thanks"... "Want some," [Margarita] asked, with salaciously puckered lips. Again I responded with a "thanks, but no thanks." (Pandemic of Lies: The Exile).
The phrase is used to turn down an unwanted offer.
So what if it incudes the same word twice? As StoneyB points out,
we are speaking language, not logic.
We often use "redundancy" to express emphasis. If Fred's wife is very pleased with a gift that Fred bought for her, is Fred going to find fault if his wife says:
'Oh, thank you, thank you, thank you. I love you. I love you. I love you'?
When Shakespeare writes:
That was the most unkindest cut of all (Julius Caesar),
are we to fault him for not sticking with either 'the most unkind cut' or 'the unkindest cut'? Why the heck was he using a pleonasm?
The function of language is to communicate meaning. To do so, we use expressions. Such expressions, when looked at in terms of logic, may contain "unneeded words." In fact, we express ourselves this way all the time.
In the first comment to this question, tchrist writes
Despite protestations to the contrary,
To which someone responds: Isn't "to the contrary" a pleonasm here? I mean, a protestation is always to the contrary, isn't it?
So what. Are we speaking language or logic?
Else, are people going to point out that when I said whole unit way above, that the word whole was unnecesary?
This use of "pleonism" is not restricted to English. In the story from Matthew 2,
When they saw the star, [the wise men] rejoiced with exceeding great joy.
This is the King James Version of the Bible. The translators didn't think to avoid a pleonism when translating from the Greek. (Although more recent translations have avoided the "dreaded pleonism.")
But the Greek faithfully records what is a frequent usage in Hebrew: that of pleonism to show emphasis.
So, in thanks but no thanks, is one of the thanks redundant? No. Not if by redundant you mean using an established phrase to express meaning in a certain context, that of turning down an unwanted or undesired offer.