Let \begin{eqnarray} X & \xrightarrow{f} & * \\ \downarrow & & \downarrow\\ Y & \xrightarrow{g} & Z \end{eqnarray} be a (strictly) commutative diagram of pointed CW-complexes with $Y$ contractible. There is an induced map $$\varphi:\operatorname{h-cof(f)}\to \operatorname{h-cof(g)}$$ on the respective homotopy cofibers of the horizontal maps $f$ and $g$. No matter what model you choose for the homotopy-cofiber, the property of $\varphi$ to be null-homotopic (i.e. the property to factorize over the point $*$ up to homotopy) is independent of this choice.
What is an example of a situation as above such that $\varphi$ is not null-homotopic?
I think I can show that $\varphi$ is null-homotopic, if $g$ was a cofibration but I presume that $\varphi$ may be non-null-homotopic, if $g$ was arbitrary. I can find an example if the upper-right corner $*$ of the initial diagram is replaced by an arbitrary contractible CW-complex, even if $g$ was a cofibration. Thank you.
You could take $Y=\text{C}X$ the cone over $X$ and define your square to be cocartesian, i.e. as the (homotopy) pushout of $\text{C}X\leftarrow X\to\ast$. Then $\varphi: \text{cof}(f)\to\text{cof}(g)$ is an isomorphism since cocartesian squares glue.