This was a question I first asked on stack exchange, here. In my head it seems like a fairly reasonable thing to ask for, but I'm not aware of any construction in the literature.

Let $p:E\rightarrow X$ be a Serre fibration over a pointed, connected CW complex $X$ with strict fibre a CW complex $F=p^{-1}(\ast)$. Given another space $F'$ and a homotopy equivalence $F\simeq F'$, is it possible to construct a Serre fibration $p':E'\rightarrow X$ with strict fibre $F'=p'^{-1}(\ast)$? If $E'$ exists, is it then possible to extend the homotopy equivalence $F\simeq F'$ to a fibre homotopy equivalence $E\simeq E'$ over $X$?

If you assume some extra structure on the fibration $p$ and the homotopy equivalence $F\simeq F'$ then things work out fairly easily, but I'm interested in the more general case.

Really I would like both questions to be taken together, since as pointed out in the comments, the projection $X\times F'\rightarrow X$ trivially satisfies the requirements of the first question alone, and such an answer is not exactly what I was looking for.

The simplicial model of Univalent Foundations (after Voevodsky)(Kapulkin–Lumsdaine 2012). Specifically, in the triangular-prism diagram in the proof of 3.4.1, take $B$ to be your $X$, $A$ to be the point of $X$, $\bar{E_2}$ to be your $E$ (so $E_2$ is the fiber $F$), and $E_1$ to be your $F'$. The extension $\bar{E_1}$ constructed there is then your desired $E'$. I don’t think it’s quite trivial to adapt the proof to the topological setting, though. $\endgroup$ – Peter LeFanu Lumsdaine Jul 19 '19 at 9:26