Good question. I think the answer is yes.

The unnamed spectral sequence is usually referred to as the isotropy spectral sequence. For a group $G$ acting on $X$ and an abelian group $A$ of coefficients as in your question, it has $E_2$-page given by the sheaf cohomology $H^p(X/G; \mathcal{H}^q)$ of the orbit space, and converges under favourable circumstances to the cohomology $H^*(EG\times_G X;A)$ of the homotopy orbit space. If $X$ is a regular $G$-complex, then the sheaf $\mathcal{H}^q$ of coefficients takes the value $H^q(G_\sigma;A)$ over a simplex $[\sigma]$ of the orbit space $X/G$. Summarising, the spectral sequence goes from cohomology of the orbit space with coefficients in the cohomology of the isotropy subgroups, to the equivariant cohomology.

This can be identified with the Leray spectral sequence of the map $EG\times_G X\to X/G$ from the homotopy orbit space to the (genuine) orbit space, given by projecting $EG$ to a point. This is not a fibration in general, but one still gets a spectral sequence, at the expense of replacing cohomology with local coefficients by sheaf cohomology. A decent reference for the Leray spectral sequence of map is the book of Bott and Tu.

Now, the Leray spectral sequence of a map works just as well for generalized cohomology theories. The details appear in the thesis of Richard Cain,

Cain, R.N., The Leray spectral sequence of a mapping for generalized cohomology, Commun. Pure Appl. Math. 24, 53-70 (1971). ZBL0205.53002.

So for a generalized cohomology theory $F^*$ and a regular $G$-complex $X$, you should get a spectral sequence with $E_2$-page $H^p(X/G; \mathcal{F}^q)$ converging to $F^*(EG\times_G X)$, where $\mathcal{F}^q$ takes the value $\mathcal{F}^q(G_\sigma)$ over a simplex $[\sigma]$ of $X/G$.

Good question. I think the answer is yes.

The unnamed spectral sequence is usually referred to as the isotropy spectral sequence. For a group $G$ acting on $X$ and an abelian group $A$ of coefficients as in your question, it has $E_2$-page given by the sheaf cohomology $H^p(X/G; \mathcal{H}^q)$ of the orbit space, and converges under favourable circumstances to the cohomology $H^*(EG\times_G X;A)$ of the homotopy orbit space. If $X$ is a regular $G$-complex, then the sheaf $\mathcal{H}^q$ of coefficients takes the value $H^q(G_\sigma;A)$ over a simplex $[\sigma]$ of the orbit space $X/G$. Summarising, the spectral sequence goes from cohomology of the orbit space with coefficients in the cohomology of the isotropy subgroups, to the equivariant cohomology.

This can be identified with the Leray spectral sequence of the map $EG\times_G X\to X/G$ from the homotopy orbit space to the (genuine) orbit space, given by projecting $EG$ to a point. This is not a fibration in general, but one still gets a spectral sequence, at the expense of replacing cohomology with local coefficients by sheaf cohomology. A decent reference for the Leray spectral sequence of map is the book of Bott and Tu.

Now, the Leray spectral sequence of a map works just as well for generalized cohomology theories. The details appear in the thesis of Richard Cain,

Cain, R.N., The Leray spectral sequence of a mapping for generalized cohomology, Commun. Pure Appl. Math. 24, 53-70 (1971). ZBL0205.53002.

So for a generalized cohomology theory $F^*$ and a regular $G$-complex $X$, you should get a spectral sequence with $E_2$-page $H^p(X/G; \mathcal{F}^q)$ converging to $F^*(EG\times_G X)$, where $\mathcal{F}^q$ takes the value $F^q(G_\sigma)$ over a simplex $[\sigma]$ of $X/G$.