Non-uniform version for the whole polynomial hierarchy The non-uniform versions of P, NP and coNP are P/poly, NP/poly and coNP/poly. 
Similarly, we can define a non-uniform version for each level in the PH.

For example: $\Sigma_2$/poly consists of problems of the form $\{x : \exists y \forall z \; C(x,y,z)\}$, where C is a circuit of polynomial size that may vary depending on the length of the input string $x$, and $y,z$ also have lengths polynomial in $x$. 

Doing this for all levels of PH, we get a non-uniform version PH/poly.

QUESTIONS:  Is there anything known about this hierarchy? Does it collapse? Or is there another name for it in the literature?
Well, sure, we know things. I think this is a pretty standard nomenclature for it. This hierarchy collapses if and only if $\mathsf{PH}$ does, exercise:


For one direction, modify the proof of Karp-Lipton to show that if $\mathsf{NP} \subseteq \mathsf{coNP}/poly$ then $\mathsf{PH}$ collapses, and observe that this result relativizes
For the other direction, see the comments by Kaveh below.