IV: Coyly, he asks: Did this come up about Coy during our read?

[David Payne]

Here's the full listing of the online OED's definitions with the etymologies in brackets and a note from me regarding the etymology of verb 1, which is what Mark K. just suggested as a fourth possibility:
 
Noun 1:
[a. Du. kooi, formerly côye, in same sense, a parallel form to MDu. couwe = MHG. kouwe, köuwe:WGer. cawia, cauwia, a. L. cavea hollow, enclosure, CAGE.] 
1. A place constructed for entrapping ducks or other wild-fowl; a DECOY. 
2. A lobster-trap. dial. 
3. = COY-DUCK. Also fig. 
4. attrib. and Comb., as coy-bird, -dog, -house, -man, -pool. Also COY-DUCK. 
 
Noun 2 (obs):
[Seems to go with COY v.2: but may be from COY v.1] 
Encouragement of an animal by clapping the hands or the like. 
 
Noun 3 (obs):
[a. F. coy ‘a sinke’, or as fosse coye ‘a priuie, jakes, house of office’, Cotgr. (lit. ‘quiet or retired ditch’.)] 
? A sink. 
 
Adj:
[a. F. coi (fem. coite) earlier quei = Pr. quetz:L. *qutus, from quitus at rest, still, quiet.] 
1. Quiet, still. Chiefly in to bear, hold, keep (oneself) coy. Obs. 
1b. of things. Obs. 
1c. Cf. a quiet hint. 
2. Not demonstrative; shyly reserved or retiring.    a. Of a person. Displaying modest backwardness or shyness (sometimes with emphasis on the displaying); not responding readily to familiar advances; now esp. of a girl or young woman. 
2b. of actions, behaviour, looks, etc. 
2c. of animals; and fig. 
2d. transf. Of a place or thing: Withdrawn from view or access, inaccessible, secluded. 
2e. Const. of or inf.: Reserved, backward. 
2f. Unwilling to commit oneself (about a matter); archly reticent or evasive. 
3. Of distant or disdainful demeanour. Also quasi-adv. Obs. 
4. to make it coy: to affect reserve, shyness, or disdain. Obs. 
5. Lascivious. (? an error; but cf. COY v.1 3.) 
 
Verb 1:
[f. COY a.: or perh. originally an aphetic form of acoy, ACCOY, OF. acoier vb.] 
1. trans. To render quiet; to calm, appease. 
2. To stroke or touch soothingly, pat, caress. 
3a. To blandish, coax, court, gain over by caresses or coaxing. Obs. 
3b. To coax, entice, allure into, from, etc. Obs. (Here, app. associated with COY n.1, DECOY v.) 
3c. intr. to coy with: to coax, blandish. 
4a. intr. To act or behave coyly; to affect shyness or reserve. Chiefly in to coy it. arch. 
4b. To disdain. Obs. rare. 
5. fig. To withdraw itself, recede into the background. 
6. trans. To disguise or slight in a demure manner. rare. 
Hence coying vbl. n.1, fondling, coaxing, blandishing. 
 
Note from Dave Payne regarding verb 1: OED for accoy says: [a. OFr. acoie-r, acoye-r to calm, appease, f. à to + coi quiet, calm:L. quit-um QUIET.] To still, calm, quiet, or appease; hence, to soothe or coax (the alarmed or shy), to tame, silence, or daunt (the forward or bold).
 
Verb 2:
[See COY n.2] 
To instigate or stir up to action. Hence coying vbl. n.2