great-

Old English great "big, tall, thick, stout, massive; coarse," from West Germanic *grauta- "coarse, thick" (source also of Old Saxon grot, Old Frisian grat, Dutch groot, German groß "great"). If the original sense was "coarse," it is perhaps from PIE root *ghreu- "to rub, grind," via the notion of "coarse grain," then "coarse," then "great;" but "the connextion is not free from difficulty" [OED].

It took over much of the sense of Middle English mickle, and itself now is largely superseded by big and large except in reference to non-material things. In the sense of "excellent, wonderful" great is attested from 1848.

Great White Way "Broadway in New York City" is from 1901, in reference to brilliant street illumination. The Great Lakes of North America so called by 1726, perhaps 1690s. 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).

"The Great War" — as, until the fall of France, the British continued to call the First World War in order to avoid admitting to themselves that they were now again engaged in a war of the same magnitude. [Arnold Toynbee, "Experiences," 1969]

Also formerly with a verb form, Old English greatian "to become enlarged," Middle English greaten "to become larger, increase, grow; become visibly pregnant," which became archaic after 17c.

a special use of grand (adj.) in genealogical compounds, originally with the sense of "a generation older than," first attested c. 1200, in Anglo-French graund dame "grandmother," also grandsire (late 13c.), from such use of Old French grand-, which perhaps is modeled on Latin avunculus magnus "great uncle." The partly-Englished grandmother, grandfather are from 15c. Other such words in European languages are formed with the adjectives for "old" or "best" (Danish bedstefar) or as diminutives or pet names (Greek pappos, Welsh taid). The French formation also is the model for such words in German and Dutch. Spanish abuelo is from Latin avus "grandfather" (from PIE *awo- "adult male relative other than the father;" see uncle), via Vulgar Latin *aviolus, a diminutive or adjective substitution for the noun.

The extension of the sense to corresponding relationships of descent, "a generation younger than" (grandson, granddaughter) is from Elizabethan times. The inherited PIE root, *nepot- "grandchild" (see nephew) has shifted to "nephew; niece" in English and other languages (Spanish nieto, nieta). Old English used suna sunu ("son's son"), dohtor sunu ("son's daughter").