If "strict fibre" just means the fibre over $*$, then you can make the construction as follows. There are two steps.
First, let $f:X\rightarrow X$ be homotopic to the identity and contract some neighborhood $N$ of $*$ onto $*$. Pulling back your fibration $p$ through $f$, you are reduced to the case where $p$ is a projection $F\times N\rightarrow N$ over $N$. Then, consider a function $g: N\rightarrow I$ whose value is $0$ on $\partial N$ and $1$ on $*$; consider the mapping cylinder $C $ of a homotopy equivalence $F\rightarrow F'$ and the projection $pr:C\rightarrow [0,1]$ with $pr^{-1}(1)=F'$; define $E'$ over $N$ as the amalgamated product of $C$ with $N$ over $G$; define $E'$ as $E$ over $(X-N)$. If you have chosen $N$ to retract by deformation on $*$, then this $E'$ will have the property you want.
Maybe there is a more elegant way to present the construction (i'm not an expert in homotopy theory nor cw-complexes) Sorry, I come back to this answer which I realize is not quite correct, because $C$ is not a Serre fibration over $I=[0,1]$, in general. Instead of $C$, you should use a Serre fibration over $I$ whose fibres over $0$ and $1$ are $F$ and $F'$ respectively; there should be a canonical construction of this kind, but right now I cannot be more precise. Hope it helps.
