I don’t know if it’s got anything to do with anything, but Chito is also the name of a cat in “Flying Witch”:
Hmm… I’ve read some of the comments, and it seems people thought it was snot, but it’s not, so she felt the need to clarify that I think.
But I’m just a beginner, so maybe someone else can help you out more. ^^;
I think this is 鼻くちょ:
I don’t know if this is cat specific or not, but when you google image search 鼻くちょ, it is mainly cats…
I asked a Japanese native, but she hasn’t responded yet…
I don’t think this is the case, that would be English grammar, but in Japanese what “someone” is thinking is written before と思う.
Yes, I forgot about the quotative と. I don’t suppose that’s one of the many dropped particles isn’t it? Lol EDIT: Wait no, brainfart. 思うwould still have to go after it.
So 鼻くちょ specifically means… kitty snot?
*follows thread with interest
I think maybe it refers to small specks of dried snot? Very weird. I can find instances of it on Twitter and in Google images, but absolutely no dictionary is proving fruitful. Once we get to the bottom of this maybe we oughta add it to Wiktionary or something lol.
Is is possible that 付いてふ is a typo? Would it make more sense as 付いてる (contraction of 付いている)?
Also, jisho.org shows ちっと (alternate form: ちと) as meaning “a little bit”.
So, “A little bit that I thought was cat snot stuck on him/her.”
?
付いてる does make a whole lot more sense, which lends that と to be the quotative for 思う. The placement of チトっ just seems weird to me though.
I was seeing it as 「お鼻くちょ付いてふと思った」being a long clause modifying チトっ, which definitely isn’t how I, as a native English speaker, would first think to write it, but 
(It still could be wrong, but I feel pretty confident)
I tried to see it that way too. Maybe the captioner is comparing their cat to the character @BlueberryPear mentioned. When I perused the search チトっ on Twitter and looked at media there was another little anime girl character that popped up a lot too.
My girlfriend said she’s never seen anyone write like this, so I wouldn’t put too much energy into trying to determine if there are any “words” you can use elsewhere. It’s either a niche inside-dialect or just a unique weirdo. Obviously she can grasp the general idea, which I believe you’ve all managed to as well, but things like using the ふ instead of る, and チトっ, etc, she’s never seen that.
I ran this sentence through Google Translate and got this:
I thought I caught your nose and suddenly thought!
I’m guessing it’s a “got your nose” thing.
'>_>
Of course, my goal here is to understand, not imitate. Besides, I mainly communicate over the internet, so who knows… Maybe I’ll run into this a lot. 
The responses I got from my Japanese friends (purely about 鼻くちょ) were:
“I think it means booger” and
“It’s used for babies/kids and maybe pets. The original word is 鼻くそ. It means booger”
maybe チト is the name of the cat in the picture?
Nope, cat’s name is サスケ.
And just to clarify to anyone who wasn’t sure why I made that face (since I can’t tell if you’re joking)
Beyond the “got your nose” game being something Japanese people won’t get because it’s not something they do with kids, it’s recommended that you don’t try to do it because the hand configuration is considered vulgar in Japan.
Oh…whoops
Well, it’s not like you were recommending it or anything, but I figured it might be interesting to know.