Hi Fiona!
The DNA test is a good place to start, as noted above. However, you noted above " He was the second cousin of my Mums Mothers husband" so the DNA is going to come up with at least distant matches regardless, if you can find relatives to match against.
Another thought is that you need to test against her "known" siblings - there may be other surprises there, as well. I only point that out, because the siblings I was raised with all turned out to be half-sibs. In that era, keeping a lack of fertility hidden, and the solution "in-house" as it were, was pretty common.
My biggest recommendation, if you go down the DNA path, is be ready for surprises.
Good luck!