Pick some countably infinite set $Y$. Call two finitely presented groups $G_1$ and $G_2$ presentation related if for some finite $X \subseteq Y$,
$G_1 \cong F\{X\} / H$ and $G_2 \cong F\{X\} / \langle H \cup \{h\} \rangle$ for some $h \in F\{X\} \setminus H$ and $H \le F\{X\}$
or
$G_1 \cong F\{X\} / H$ and $G_2 \cong F\{X \cup \{y\}\} / i(H)$ for some $y \in Y$ and $H \le F\{X\}$ where $i: F\{X\} \mapsto F\{X \cup \{x\}\}$ is the natural injection induced by the inclusion of sets.
Also, as two isomorphic groups always have identical presentations, it makes sense to define that two isomorphic groups are trivially presentation related.
We now have a graph on the finitely presented groups where there is an edge between two nodes/groups if those groups are (non-trivially) presentation related.
This also gives a metric by shortest paths / geodesic distance.
Is this construction useful? (Answer to comment: ) Or more directly, is this construction known and/or used in research mathematics?
Given the problem of putting a natural metric on the class of finitely presented groups, I believe many mathematicians would come up with something substantially similar to this.