0 \to M' \to M'' \to M \to 0
where you know the graded betti numbers for M' and M''. In this case, the
mapping cone can occasionally give you information about the graded betti
numbers of M.
Similarly, if you can find a short exact sequence
0 \to M' \to M \to M'' \to 0,
where again you know minimal free resolutions for M' and M'', then it is
sometimes possible to garner information about a minimal resolution of M
by reading the mapping cone "backwards".
I plan to give several examples of this technique, and discuss its limitations.