Thank you very much to everybody, who answered me! Here are few observations, concerning your answers.
The answer by Johannes Ebert is the closest to what is now on my mind. His is the answer, which I guessed would and should be right (i.e. The $\Omega X$-Borel-equivariant homology of $Y$ is the homology of $E$), even though it is a bit too simple to spark interest, of course. But at present I just need a reference to a paper in which such construction is dicussed: there are few moments which are not evident for me, e.g. it is not quite clear for me now, what is the proper way to look at the universal space $E\Omega X$, since $\Omega X$ is not a group, but only a homotopy group, etc. (clearly, it should be equivalent to the based paths space, but I need an explicit construction, similar to the bar-resolution, or something like this, which would fit arbitrary good H-space). It seems from what Johannes says, that these are well-known things, so, dear Johannes, if you have a reference to any paper, in which such questions are discussed, please, let me know.
And of course, I would like to have a deeper understanding of such equivariant theories in general, i.e. in the full generality of the local system approach, so, my special thanks to Tyler Lawson and David Ben-Zvi for an extensive list of possible constructions! In effect, my interest is caused by an attempt to understand the topological counterpart of various equivariant theories with coefficients, in the case when the group is replaced by a d.g. Hopf algebra or something equivalent, so this is what I really need!