I'm not a grammar expert and English is not my first language but I think I used to say this before and I just ended up taking out the "what it is" and changed it for the thing I'm trying to remember:
There's a thing that I don't know the name of
Or
There's a thing that I don't know how to describe
Or
There's a thing whose purpose is a mystery to me
Is that what you're refering to? Sorry if it's not. I don't think any of the first three examples are correct, or at least they sound really weird to me.
Please do correct me if there's an English mayor somewhere though!
To clarify - I think your proposed grammar is valid but the phrasing is uncommon. It’s not a phrase I would expect to hear. Though I would understand the gist of what you’re expressing.
This is a great question, and it led me down a bit of a rabbit hole. This kind of clause is called a Gapless Relative Clause. The sentence could be written as you have it, or with "I don't know what it is" - the "it" is called the Resumptive Pronoun which are "common in spoken English but are officially ungrammatical".
The Wikipedia article has a similar example:
In other cases, the resumptive pronoun is used to work around a syntactic constraint:
They have a billion dollars of inventory that they don't know where it is.
In this example, the word it occurs as part of a wh-island. Attempting to extract it gives an unacceptable result:
*They have a billion dollars of inventory that they don't know where ___ is.
"Resumptives are non-standard, but in such cases they're much better than their gapped counterparts, which people usually find incomprehensible, or at least very hard to comprehend."
So basically, your original sentence is "unacceptable"/"incomprehensible", but adding "it" would be grammatically incorrect but easier to understand. Best bet is probably to totally rephrase the sentence as others have suggested.
I'm struggling to think of a context where you'd say this where you couldn't just say "I don't know (about) that thing" or "there's a thing I don't know".
To keep as much of your original sentence as possible, "There's a thing, (I don't know what it is,) and...."
Basically the only way to smush those two concepts together in the order you'd like is if one is basically an interjection to an ongoing sentence containing the other. In that case, you wouldn't use any connecting words at all.
If you wanted a single sentence, many other commenters have already given great suggestions.