Palin, a student of ancient Norse languages (as well as Plato, Pascal and Aristotle), tells us Trig was named because in ancient Norse, Trig means "true" or "loyal". This is how the Palins first explained the name in August 2008:

The name Trig is a Norse word meaning "true" and "brave victory," Leighow said. Paxson is an area of Alaska that both Palin and her husband, Todd, feel is "one of the most beautiful spots in Alaska," she added.

Palin earlier joked about naming the baby Van Palin after the '80s rock group Van Halen.

Some accounts also say that Trig is a family name, although Palin never actually names anyone in her extended family by that name. But a reader suggests something else: that Down Syndrome is referred to medically as Trisomy-12 or Trisomy-G.
Could Trig's name actually be a reference to his special needs: Tri-G, a common medical shorthand for Down Syndrome?