My grandson was in a round robin baseball tournament (in which every team played every other), and someone asked me why it was called “round robin.” I didn’t know, and it turns out nobody really knows, but what astonished me when I checked the OED (where the entry was updated in March 2011) was the sheer variety of things to which the phrase has been applied. The first branch of senses falls under the general rubric “circular objects”; it is first attested as “a disparaging name for the consecrated Host at the Eucharist” (?1548 tr. J. Calvin Faythfvl Treat. Sacrament Pref. sig. Aiii, “Certeyne fonde talkers..applye to thys mooste holy Sacramente, names of despitte and reproche, as to cal it Jake in the boxe, & rounde Roben, and suche other not onely fonde but also blasphemouse names”), then as a type of small ruff (1642 J. Taylor Apol. for Private Preaching sig. A3, “King Arthur of Bradley, and his four hundred forty sixe Elders of the Round Table, the first men that ever wore Round-Robins”), as a rim or plate on a carriage axle (1795 W. Felton Treat. Carriages II. 232 (Gloss.), “Round Robbins, broad rims fixed to the ends of the axletree bed..for preventing dirt falling in to injure the arms of the axletree”), as a loop of leather or rubber through which passes a pole, spring, or other part of a carriage and by which it is suspended, and (in Devon) as a small pancake.
Branch II is oddly labeled “A person, fish, or plant,” encompassing a sense “Applied to a man (in various allusive uses)” (1591 R. Greene Notable Discouery of Coosenage f. 8, “There in faith round Robin his deputy, would make them like wretches, feele the waight of his heauiest fetters”); several kinds of fish—”a freshwater sunfish of eastern North America, either the pumpkinseed, Lepomis gibbosus, or the redbreast sunfish, L. auritus,” “Any of various marine fishes of the genus Decapterus (family Carangidae), characterized by a fusiform body almost circular in cross-section; esp. the round scad, D. punctatus, of Atlantic coastal waters, widely fished for food and as bait,” and (in Cornwall) “the anglerfish Lophius piscatorius“; and (in the southwest of England) “any of several hedgerow plants with pink flowers, esp. the red campion, Silene dioica.”
Branch III, “Other uses,” includes “a document (esp. one embodying a complaint, remonstrance, or request) having the names of the signatories arranged in a circle so as to disguise the order in which they have signed” (1698 C. Young in High Court of Admiralty Exam. & Answers (P.R.O.: HCA 13/81) f. 679v, “Some of them drew up a paper commonly called a Round Robin, and signed the same whereby they intimated that if the Captaine would not give them leave to goe a shore, they would take leave”), “a letter copied and sent to several recipients” (1871 Monthly Packet June 591 “Shall I send a round robin, telling them that you would not have time to write books if you answered all their questions?”), “a letter, piece of writing, etc., sent around the members of a group, and added to by each recipient in turn. Cf. chain letter” (1885 A. Baker More Half-hours with my Girls 204 “How can I write to you all? I must write a kind of round robin, to be sent to each in turn”), “a tournament in which every player or team competes once with each of the others” (1894 N.-Y. Times 28 Sept. 3/5 “A well-known player has proposed that the four players who reach the semifinals play a kind of ‘round-robin’ tournament”), “a group activity consisting of successive participation from each member of the group” (1915 Atlanta Constit. 7 Nov. 4 f/6 “Chairman of the entertainment committee, R. S. Wessels, is going to invoke what he says is a ‘Round Robin’ method of introducing them. Every man must rise in his place and introduce the member next to him.”), and “a wager consisting of three selections in different events, which are combined in various ways to produce ten individual bets (three doubles and a treble, and three pairs of ‘up-and-down’ bets)” (1944 G. Panetta We ride White Donkey 110 “‘Give me Fogoso, Doubt Not, and Chop Suey’ I said, ‘two dollars each in a three-way round robin.’”).
The etymology basically throws up its hands:
The senses at branch I. all denote things which are circular, and those at branch II. are connected with simplex uses of robin n.1, while the senses at branch III. denote something which goes round or around. However, in no case is it entirely clear what determined the collocation of the words round adj. and robin n.1, except for alliteration. It is probable that the phrase became well established in one or more uses (perhaps originally sense 1, if the chronology implied by the attestations is correct), and then became extended to other things which were round, or went around, or were similar to things otherwise referred to as robin n.1 However, what the original motivation may have been remains uncertain.
In other words, “What can we tell you? People like to say it because it sounds good.”