First, I am a little surprised that Illinois tax is more than NY tax, but I guess at the lowest levels it is - Illinois is 4.95% while NY is 4%, 4.5%, 5.25%, 5.9%, 6.21%, 6.49%, 6.85% and 8.82%.
No matter, this is one of those undefined areas. When the HSA contribution is tied to state-source income, the answer is easy. When the contribution is not directly tied to state-source income, then there are generally no specific instructions. Here I would fall back to where you were living at the time of the contribution.
You are paying the contribution out of an amorphous bucket of money. Who is to say if the contribution dollars came from NY income or IL income?
If you choose to take the stand - lacking any clear instructions from either state - that you want to allocate the contribution between the states, just make sure that you document what you did and why.
And if one state ever questions your choice and assesses additional tax on you, well, you'll be able to file an amended return for the other state to get the money back (more or less).
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