I would say Serena for the women. No women has ever won as many big slam matches from the brink of defeat or saved as many match points against her. King, Evert, Graf, and Seles would be my honorable mentions with King probably taking runner up. I might have picked Seles but her poor comeback from the stabbing was not the stuff of the toughest competitor.
For men I would say Connors. Honorable mentions to Nadal, Sampras, Muster, Chang, and Hewitt.