A recent question prompted the answer of "gadabout" - which made me think of "gadfly".
Are the two related? If so, which gave rise to the other; or, did they occur contemporaneously? What's a "gad"?
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Summary (paraphrased from Etymonline): Gadfly probably comes from gad (n), a goad, but "the sense is entangled with gad (v) 'rove about'". Gadabout comes from gad (v) plus about. The noun, gad, is older than the verb, gad (from gadden); both are older than gadfly. (The verb gad may perhaps be derived from the noun gad.) Gadabout is comparatively recent. More complete etymologies follow.
gadfly (n.) also gad-fly, 1620s, "fly which bites cattle," probably from gad (n.) "goad, metal rod," here in the sense of "stinger;" but the sense is entangled with gad (v.) "rove about" (on the notion, perhaps, of the insect's power of flight or of the restlessness of animals plagued by them), and another early meaning of gadfly was "someone who likes to go about, often stopping here and there" (1610s). Sense of "one who irritates another" is from 1640s
gad (n) c. 1300, "a goad, sharp pointed stick to drive oxen, etc.;" c. 1400, "sharp-pointed metal spike," from Old Norse gaddr "spike, nail," from Proto-Germanic *gadaz "pointed stick"......
gad (v) mid-15c., gadden, "go quickly, hurry," of uncertain origin, perhaps from gad (n.) "sharp stick for driving oxen" on the notion of moving as animals do when being driven by a gad.
gadabout (n.) "one who gads or walks idly about, especially from motives of curiosity or gossip" [Century Dictionary], 1830; see gad (v.) + about (adv.). As an adjective from 1817.....
(Emphasis added to definitions for clarity.)
Ernest Weekley, An Etymological Dictionary of Modern English (1921) gives no indication that the terms gadabout and gadfly use gad- in an etymologically shared sense. Here are Weekley's entries for the the relevant terms (I omit his coverage of gad in the sense of "God," as in gadzooks):
gad1. Spike. O[ld] N[orse] gaddr, spike, nail, associated with the unrelated A[nglo-]S[axon] gād, goad. Hence gad-fly. With quot[ation] [see below] cf. spur of the moment. [Example:] All this was done/Upon the gad (Lear, i. 2).
gad3. To wander aimlessly. Back-formation from obs[olete] gadling, A[nglo-]S[axon] gœdeling, comrade, cogn[ate] with gather. In M[iddle] E[nglish] this meant base fellow and in 16 cent[ury] vagabond. For gad-about cf. earlier gadder about (N[ew] E[nglish] D[ictionary] 1568).
Unlike Etymology Online, Weekley sees an association between the Old Norse word for spike and the Anglo-Saxon word for goad, but no crossover between the those senses of gad and gad as a short form of gadling meaning aimless wandering. In reaching this conclusion, Weekley seems to break intentionally from Walter Skeat, An Etymological Dictionary of the English Language (1888), whose work he clearly consulted. Skeat has two entries for gad:
GAD (1) a wedge of steel, goad (Scand[inavian]) 'A gad of steel;' Titus Andron[icus] iv. 1. 103. Also 'upon the gad,' i.e. upon the goad, suddenly; K[ing] Lear, i. 2. 26. 'Gadde of steele, quarreau dacier;' Palsgrave. M[iddle] E[nglish] gad, a goad or whip; 'bondemen with her gaddes' = husbandmen with their goads or whips; Havelok, 1016. — Icel[andic] gaddr (for gasdr), a goad, spike, sting, cognate with E. goad, yard. See Goad, Yard. Der[ived term] gad-fly, i.e. sting-fly; and see gad (2).
GAD (2), to ramble idly. (Scand[inavian]) 'Where have you been gadding?' Romeo [and Juliet], iv. 2. 16. 'Gadde abrode, vagari;' Levins, 7. 47. The orig[inal] sense was to drive, or drive about. — Icel[andic] gaddr, a goad. See above. I see no connection with M[iddle] E[nglish] gadeling, an associate, fir which see Gather.
So if you follow Skeat (1888), gadfly and gadabout share a Scandinavian ancestor equivalent to Icelandic gaddr; but if you follow Weekley (1921), the two words derive from fundamentally unrelated precursors to the coincidentally same syllable gad.
As a side-note, Thomas Wright, A Dictionary of Obsolete and Provincial English (1857) lists nine meanings of gad (not including "to wander aimlessly"), as well as meanings for gad-about ("A rambler"), gader ("To gather"), gading/gadding ("A going about; a pilgrimmage"), and gadling ("A worthless vagabond").
"Late 16th century: from gad1, or obsolete gad ‘goad, spike’, from Old Norse gaddr, of Germanic origin; related to yard1." -- gadfly. Dictionary.com. Dictionary.com Unabridged. Random House, Inc. http://www.dictionary.com/browse/gadfly (accessed: April 30, 2016).
Is the etymology give for the 'gad' in 'gadfly'. I would suggest that to goad or to be spiky in countenance is very similar to what a gadabout might do, so I would suggest that some common ancestry is likely, though I could find not sources for 'gadfly' regarding the word original. At a guess: yes, they do have common origin... probably.