The term demigod referred to a minor deity, often the offspring of a god and a mortal.
According a prediction made in the Takarian epic poem, the Song of the Sages, two demigods, known as the Holy Sages, were to arrive from the sky. These two, identified as the Greater and Lesser Sage, would then rule over the people as benevolent protectors. (VOY: "False Profits")
The term could also refer to hybrids descended from Greek gods and mortal species, though it was not used officially by Starfleet. One example was Olly, the granddaughter of Zeus, who served in Starfleet. Olly had inherited some powers from her ancestors, specifically the ability to absorb energy and form physical lightning bolts. However, Olly hated her powers and as a result never learned to properly control them, resulting in them regularly short-circuiting electronic systems around her. (LD: "Of Gods and Angles")
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Some figures from mythology qualify as demigods, although none were identified as such on screen. These include Achilles, Asclepius, Gilgamesh, Hercules (though he was elevated to true godhood after he died), and Romulus and Remus.
Also, by definition, Benjamin Sisko would qualify, as his mother was possessed by aliens worshiped as gods at time of his birth. By the same definition, Jesus Christ would also qualify.
Furthermore, a deleted scene from "Who Mourns for Adonais?" would have revealed that mortal Enterprise crewmember Carolyn Palamas was pregnant with the god Apollo's child at the end of the story.