The act (or physical remnants from the act of) taking a large and masculine defication, or shit.
Whoa, I'll be right back, I gotta take a man dump.
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A dump so large, or so necessary, and so relieving as to have you walk away from the can smiling and saying "OMG, what a great dump, I feel like a new man." Hence, a "new man dump."
After eating at Taco Joe's last night, then a few drinks at the bar afterwards, plus a midnight snack, I took a New Man Dump this morning. I felt so good I am still smiling.