You said "a male half-second cousin once removed of mine (who shares this same g-grandfather)" took the DNA test, and that you have a common surname in that g-grandfather.
I'm not sure what you meant about your cousin's relationship to the great-grandfather. Was your male cousin who took the Y-DNA test a male of the SAME surname as your great-grandfather, in a straight male-descent line? That's what the Y-DNA test is for. It's not just that you should have the same gr-grandfather, such as John Smith, and each of you is descended from a child of John, male or female. The Y-DNA participant has to be the son of a son of a son of a son, all carrying the same surname (and usually, but not always, the same DNA patterns).
DNA testing has the potential to reveal non-parental events, so that the surname you're expecting to match is NOT the surname (DNA) brought down through your male surname lineage. My Armstrongs from Northern Ireland are in fact pure Irwin DNA, due to a non-parental event at some point before about 1765 (we think at least a century or more earlier).
I'm sure others on this forum will have more to tell you about DNA, but you're welcome to write me off the board and tell me in particular what the gr-grandfather's surname was, and what your stumbling-blocks were in not finding his parents. Was he a resident of the USA, and in what state? What records DO you find on him? There might be a clue that indicates further research avenues for you.