From one of the threads above:
**This is a devotion of reparation to the Immaculate Heart of Mary, done to honour Our Lady’s sorrows ...**
Of course, part of Friday, Saturday, and part of Sunday were the days that Christ was in the tomb.
And from another one of the threads linked above:
**It may come as some surprise that this devotion requested by Heaven was not absolutely new. It fit precisely into the long tradition of Catholic piety that, having devoted Fridays to the remembrance of the Passion of Jesus Christ and to honoring His Sacred Heart, found it very natural to devote Saturdays to His Most Holy Mother.**