to conquer overwhelmingly: to win far beyond an ordinary victory, to be more than a conqueror
This is a strengthened form of the ordinary verb for winning or conquering (νικάω). The prefix ("hyper," the same root behind the English "hyper-") pushes the base verb to its highest pitch: not merely to win, but to win decisively, to conquer beyond conquering. Paul uses the word once, at the climax of Romans 8, and he may well have coined it himself; he had a marked fondness for piling this prefix onto words to reach for something the plain word could not hold.
The force is genuinely intensive, not decorative. Every major lexicon agrees the word means "an overwhelming, surpassing victory," which is why English versions reach for a phrase ("more than conquerors," "win an overwhelming victory") rather than a single word. The one place scholars differ is over what kind of "more" Paul intends, and that is a question about his theology, not about the meaning of the word.
What makes it striking is where Paul places it. He has just quoted a psalm describing believers as sheep marked for slaughter, and into that picture of visible defeat he drops this word for surpassing triumph. The victory is not won by escaping the hardships but in the very midst of them, and its ground is not the believers' own strength but the love of the one who died for them.