The story
The organization began in the fall of 1898, when two traveling salesmen, John H. Nicholson of Janesville, Wisconsin, and Samuel E. Hill of Beloit, Wisconsin, met in a hotel room they shared at the Central House Hotel in Boscobel, Wisconsin, and discussed the formation of an association. In May 1899, the two met again in Beaver Dam, Wisconsin, where they decided the goal of their association would be to unite traveling salesmen for evangelism. In July 1899, Nicholson, Hill, and Will J. Knights met at the YMCA in Janesville. Two of them continued with the distribution of the Bibles. Gideons began distributing free Bibles, the work they are most known for, in 1908, when the first Bibles were placed in the rooms of the Superior Hotel in Superior, Montana.
The organization describes its connection to the story of Gideon:
"Gideon was a man who was willing to do exactly what God wanted him to do, regardless of his own judgment as to the plans or results. Humility, faith, and obedience were his great elements of character. This is the standard that The Gideons International is trying to establish in all its members, each man to be ready to do God's will at any time, at any place, and in any way that the Holy Spirit leads."
In keeping with this symbolism, the symbol of the Gideons is a two-handled pitcher and torch recalling Gideon's victory over the Midianites as described in Judges 7.
From Wikipedia’s article on The Gideons International, shared under CC BY-SA 4.0.4