Loosely based on The Tale of the Heike. The storyline of the game follows roughly the third section of the Tale of the Heike which deals with Yoshitsune’s rise against the Taira clan while the character’s early life is based on information from the Gikeiki. Many of the characters in the game also appear in the Tale of the Heike.
The game’s storyline and time period corresponds with the Genpei War where the Minamoto and Taira clans fought for dominance. The Minamoto won and established the Kamakura shogunate ushering in the Kamakura period. This is considered the beginning of the era of rule by the samurai and hence the reference in the title.
Source: Wikipedia, "Genji: Dawn of the Samurai," available under the CC-BY-SA License.