What specific effect(s) does the borderlands 2 weapon modifier "increases weapon stability" have on a gun?

Solution 1:

According to the Borderlands Wiki:

Another important thing to take into account when choosing what sniper rifle is going to serve the best is an invisible quality known as stability. Stability only becomes a factor when sighting a sniper rifle, and it is measured in how fast the weapon stops swaying when first sighted, between shots, and when the user takes damage.