Old English great "big, tall, thick, stout; coarse," from West Germanic *grautaz "coarse, thick" (cf. Old Saxon grot, Old Frisian grat, Dutch groot, German groß "great").
Said to have meant originally "big in size, coarse," and, if so, perhaps from PIE root *ghreu- "to rub, grind." It took over much of the sense of Middle English mickle, and is now largely superseded by big and large except for non-material things.
As a prefix to terms denoting "kinship one degree further removed" (early 15c., earliest attested use is in great uncle) it is from the similar use of French grand, itself used as the equivalent of Latin magnus. An Old English way of saying "great-grandfather" was þridda fæder, literally "third father."
In the sense of "excellent, wonderful" great is attested from 1848. Great White Way "Broadway in New York City" is from 1901. Great Spirit "high deity of the North American Indians," 1703, originally translates Ojibwa kitchi manitou. The Great War originally (1887) referred to the Napoleonic Wars, later (1914) to what we now call World War I (see world).
Also formerly with a verb form, Old English greatian, Middle English greaten "to become larger, increase, grow; become visibly pregnant," which became archaic after 17c.