Here's a further generalization of the generalization from YCor's answer.

Claim: Let $\mathcal K$ be a locally $\kappa$-presentable category containing a coequalizer diagram $A \rightrightarrows B \twoheadrightarrow C$ where $B,C$ are $\kappa$-presentable. Then there exists a coequalizer diagram $A' \rightrightarrows B \twoheadrightarrow C$ with $A'$ $\kappa$-presentable and a map $A' \to A$ making the obvious diagram commute.

It follows, for example, that if $B$ is any finitely-presentable module and $C$ is any finitely-presentable quotient, then the kernel is finitely-generated (we need not assume that $B$ is free). The result will still be most useful in "algebraic" categories $\mathcal K$ (the context YCor described already discussed) where there is a good supply of coequalizer diagrams. Nevertheless, it's true in this more general context.

Proof: Write $A = \varinjlim A_i$ as the $\kappa$-filtered colimit of $\kappa$-presentable objects. We have resulting coequalizer diagrams $A_i \rightrightarrows B \twoheadrightarrow C_i$, and $C = \varinjlim_i C_i$. In fact, the isomorphism $C = \varinjlim_i C_i$ lifts to the coslice category $\mathcal K_{B/}$. Because $C$ is finitely-presentable (even regarded as an object of $\mathcal K_{B/}$), the identity map $C = C$ factors through some stage of the colimit, $C \rightarrowtail C_i \twoheadrightarrow C$ This is true in the coslice category, so that $(B \twoheadrightarrow C_i) = (B \twoheadrightarrow C \rightarrowtail C_i)$. Because this map is epi, it follows that $C \rightarrowtail C_i$ is epi. But it is also split mono, and hence iso. Thus $C = C_i$ is the coequalizer of $A_i \rightrightarrows B$.

Here's a further generalization of the generalization from YCor's answer.

Claim: Let $\mathcal K$ be a locally $\kappa$-presentable category containing a coequalizer diagram $A \rightrightarrows B \twoheadrightarrow C$ where $B,C$ are $\kappa$-presentable. Then there exists a coequalizer diagram $A' \rightrightarrows B \twoheadrightarrow C$ with $A'$ $\kappa$-presentable and a map $A' \to A$ making the obvious diagram commute.

It follows, for example, that if $B$ is any finitely-presentable module and $C$ is any finitely-presentable quotient, then the kernel is finitely-generated (we need not assume that $B$ is free). The result will still be most useful in "algebraic" categories $\mathcal K$ (the context YCor described already discussed) where there is a good supply of coequalizer diagrams. Nevertheless, it's true in this more general context.

Proof: Write $A = \varinjlim A_i$ as the $\kappa$-filtered colimit of $\kappa$-presentable objects. We have resulting coequalizer diagrams $A_i \rightrightarrows B \twoheadrightarrow C_i$, and $C = \varinjlim_i C_i$. In fact, the isomorphism $C = \varinjlim_i C_i$ lifts to the coslice category $\mathcal K_{B/}$. Because $C$ is $\kappa$-presentable (even regarded as an object of $\mathcal K_{B/}$), the identity map $C = C$ factors through some stage of the colimit, $C \rightarrowtail C_i \twoheadrightarrow C$ This is true in the coslice category, so that $(B \twoheadrightarrow C_i) = (B \twoheadrightarrow C \rightarrowtail C_i)$. Because this map is epi, it follows that $C \rightarrowtail C_i$ is epi. But it is also split mono, and hence iso. Thus $C = C_i$ is the coequalizer of $A_i \rightrightarrows B$.