Since I got no responses to this question at Stack Exchange, please let me try my luck here.
Call a continuous map $\pi:E\to B$ between CW complexes a homotopy fiber bundle if for any $x$ in the image of $\pi$, there is an open neighbourhood $U\subset B$ of $x$ and homotopy equivalence $\pi^{-1}(U)→U\times F$ over $U$.
I don't know if this has a different name in the literature or even if it is reasonable. Replacing ''homotopy equivalence'' by ''homeomorphism'' should be the definition of an ordinary fiber bundle.
How relates a ''homotopy fiber bundle'' to the notion of a Serre fibration?
At least both properties imply that the fibers over connected components are all weakly homotopy equivalent.
Best Answer
There are many "homotopy fiber bundles" which are not fibrations. For example take any homotopy equivalence $E \to B$ that is not a fibration, then it is a "homotopy fiber bundle" with a one-point fiber.
On the other hand the other implication is (almost) true. The following works for Hurewicz fibrations. I don't whether it is true that a fiber of a Serre fibration between CW-complexes has a homotopy type of a CW-complex. If this is the case, then the proof works also for Serre fibrations.
Let $\pi : E \to B$ be a Hurewicz fibration between CW-complexes and $x \in B$. CW-complexes are locally contractible, so there is a contractible neighborhood $U$ of $x$. Let $\pi_U : E_U \to U$ be the restriction of $\pi$ to $U$. If $E_x$ is the fiber of $\pi$ at $x$, then the inclusion $E_x \to E_U$ is a pullback of the inclusion $\{x\} \to U$ along a Hurewicz fibration, so it is a homotopy equivalence and admits a homotopy inverse $f : E_U \to E_x$. Thus $(\pi_U, f) : E_U \to U \times E_x$ is a homotopy equivalence over $U$.