What is the record for the most match points saved by a tournament winner throughout the tournament? (What is the record for ATP tournaments, WTA tournaments, Grand Slams?)
We have a similar question about most match points saved in one match: Maximum match point saved in a tennis match? From the players mentioned there, Adriano Panatta saved 11 match points in his first round match at 1976 Italian Open and also won the whole tournament. The tournament was a part of the Grand Prix tennis circuit, which was a predecessor of the ATP World Tour.
This article on the ATP Tour website and also this tweet say that Kližan's 8 match points is the most at an ATP tournament since 2001. But it is not mentioned who saved eight or more match points and still won the title in 2001.
EDIT2: On WTA tournaments, the highest number I have seen is 8 match points saved by Andrea Petkovic in 2015 Antwerp. WTA: Petkovic Caps Dream Week In Antwerp (Wayback Machine) At 2001 Miami, Jennifer Capriati lost in the final to Venus Williams, despite having eight championship points.