No, I concur with Chefguy. If “It” is being used as a proper noun, I’d argue “It’s” is the proper possessive. You would say “Cousin It’s costume,” wouldn’t you? (Well, at least if they spelled it that way. Looks like it’s “Itt” not “It.”) I suppose it depends on how you’re analyzing “It” in this case.
I would recommend everyone read a GD thread by the subject of this pitting on why there are so few black Olympic swimmers. I don’t think you’ll bother attempting to give him evidence again.
Because I suck at the Pit (although support the OP on this topic), let me just get back to this linguistic sidetrack. What I mean by this is if you’re analyzing “It” as a true proper noun. For example, if you have somebody who names their child It, say It Sanders, or if somebody has the family name It, you would, of course, use “It’s.” After all, if you had the surname She or He (both real surnames), you wouldn’t change it to Hers or His.
Now, in the OP’s case, it may be more proper to say that “It” is a pseudo proper noun (a term I’m pretty sure does not exist linguistically, but bear with me.) The capitalization is for effect, but it does not change the pronoun nature of the word, much like the capitalization of He or His in reference to God. Just depends on how you parse it. I could go either way on this one.
What pulykamell said. “It” in this case is the name of a person, hence the capitalization, so normal grammar rules for “its/it’s” don’t apply. However, this gives birth to a contender for the shortest sentence in English: when asked to whom something belongs, the answer is “It’s It’s”.
The Knights of Ni are crumpled on the floor in pain after the turn this thread took.
Where are the grammar Nazis when you need them?
I actually thought about this while writing the OP and didn’t reach any conclusions.
But in the spirit of It, “It’s a straw man that will never happen, you punk ass It-ists”.
Uh oh. Oh no! Oh me. Oh my. No no.
So go!
Did you hear me? Go!
I apologize for the accusations about lying. It was uncalled for and unwaranted.
I accept your apology. We’ll talk about redundancies another time.
Apology accepted. Thank you.
Yeah, well, I want to know if It put the lotion on Its skin! :mad:
*Or did It get the hose again? :dubious: 
Thanks, Monkey. That’s very decent of you.
Hmm - I was thinking more along the lines of arse licking (as in blowing someone that’s mooning)…and was thinking that if the monkey in question was licking the wrong arses it would explain why it was so ignorant and foul mouthed.
If it was an obvious redundancy, then calling it a redundancy is a redundant.
Would
It’s it isn’t it, is it It?
be correct?
I endorse this Pitting.
The things that makes me angriest is seeing someone who’s wrong, and who’s been shown the proof of his error, respond by blindly denying it (“I don’t care what your cite says, I’m right,” or brushing it off: “OK, I’m wrong, whatever, who cares.”) If the evidence says you’re wrong, then admit it, and modify your argument or give it up.
Ooh, the treasured Bricker endorsement, I’m gonna swoon. 
You would be wrong.
And, in keeping with the spirit of things: Goddamned fucking shit. Helldamnfuckwhore.
I’m totally on board with this pitting. After all that and dozens of people saying so, he just slinked off, the dirty little rat coward. He never came back.
I, too, have heard that argument, btw.
No, I wouldn’t. It depends on how the question is being asked. If “It” is a real honest-to-goodness name, then, yes, it’s “It’s.” Tell me, how would you indicate possession of the various names in my post?
In the particular case in the OP, I’d probably lean more towards “Its”, for the reasons I stated above (I still think it acts as a pronoun, despite the capitalization), but if someone wants to parse it as a proper name (for example, as an actual nickname), “It’s” is correct. Like I said, I can go either way on this particular case. However, in a case where somebody’s forename or surname is “It,” there is no ambiguity, “It’s” is correct.