One of the clearest early expositions of the meaning of the expression appears in a letter from a St. Louis, Missouri, hotel, dated November 18, 1835, printed in the [Springfield, Illinois] Sangamo Journal (November 28, 1835):
I wish you to watch the movements of A . G . H.--You know he never tires. What he lacks in talents, be makes up by industry and perseverance ; he is hard to head in a grocery or upon a corner[.] I wish you to keep an eye upon Stuart's big lion of the north. I fear him more than any man in my district ; for he supports four Presidents, and I only support one,—which gives him a great advantage over me. As for General Maxwell, I can spin him into wrapping thread, and wind him round my little finger. A good situation will place him on the shelf, or at least the promise of one.
To similar effect 30 years later is Charles Dickens, "Mrs. Lirriper's Legacy" (1864), in Christmas Stories from "Household Words" and "All the Year Round" (1868):
Not you see but what I knew I could draw the Major out like thread and wind him round my finger on most subjects and perhaps even on that if I was to set myself to it, but him and me had so often belied Miss Wozenham to one another that I was shamefaced, and I knew she had offended his pride and never mine, and likewise I felt timid that that Rairyganoo girl might make things awkward.
The image is thus of a tailor or seamstress drawing out a thread from his or her sewing work and wrapping that thread around a finger. This seems likely to reflect the way people in the eighteenth and nineteenth centuries imagined the expression, given that sewing was a standard task during that era in households throughout the English-speaking world. Moreover, on the frontiers of North America, handiness with needle and thread was by no means a skill reserved for women only.
The earliest wordings of the descriptive image of figuratively wrapping, twisting, twining, twirling, or turning a person around one's finger seems to have occurred in the 1700s. In addition to the February 1780 Town and Country Magazine instance that Mari-LouA cites in her answer (which my Google Books searches did not turn up), the following instances are among the earliest.
Earliest of all is this one from a letter from William Stephens to Mr. Martyn dated November 25, 1743, in The Colonial Records of the State of Georgia, 1742–1745, volume 24 (1915):
The Scandalous reflection thrown at Mr. Parker, of his being drunk, is a vile falshood : for I do aver yt [that] there was not the least appearance of it during the time of his being in Town, wch was several days. Whether the Recorder was the Author of that Report, or Remington himself, remains yet a Question ; the Recorder Alledging yt he never said so to Remington, nor that Watson could wind Parker round his finger; & yt he was ready to swear twas all false : ...
(This match from 1743 and the 1809 match below from Mason Weems appear in Bartlett Whiting, Early American Proverbs and Proverbial Phrases (1977), the source from which I learned of them.)
From Hannah Hewit, Hannah Hewit: Or, The Female Crusoe. Being the History of a Woman ... who ... was Cast Away in the Grosvenor East-Indiaman: and Became for Three Years the Sole Inhabitant of an Island, in the South Seas (1792) [combined snippets]:
No man ever possessed these diabolical qualities in a stronger degree than Thomas Sourby; and, of course, no man could be more calculated to wind round his finger such an easy, vain, credulous creature as John Hewit.
From Charles Kemble, Plot and Counterplot: Or, The Portrait of Michael Cervantes; a Farce in Two Acts (1808):
Fabio. Oh, sir, you have no notion what an amiable creature I am when I set about it. Well, sir, this widow has been some years in the service of Hernandez's sister, Donna Clara, and, by a little of my advice, contrives to turn her round her finger. She told me the other day, that Donna Lorenza had requested her mistress to inform herself privately of the character and prospects of your rival ...
From Mason Weems The Life of General Francis Marion: A Celebrated Partisan Officer in the Revolutionary War (1809):
"I was never at a loss before," said he [a lieutenant serving under Marion], "to manage all other officers that were ever set over me. As for our colonel, (meaning Moultrie) he is a fine, honest good-natured old buck. But I can wind him round my finger like a pack-thread. But as for the stern, keen-eyed Marion, I dread him."
From Belle Assemblée: Or, Court and Fashionable Magazine (1821) [combined snippets]:
A gentleman, at a fashionable party, being asked by a lady his opinion of a beautiful ring she wore, in which was a very small miniature, and most striking likeness of her husband, observed, that he was no great judge of painting, and having seen Lord ——— but once, he was hardly competent to pronounce on the likeness; nevertheless, he was happy to see her Ladyship had a husband that she could turn round her finger.
Conclusion
The earliest written instance that I've been able to confirm of figuratively wrapping someone around one's finger is from 1743—an instance in which a letter writer in Georgia denies that "Watson could wind Parker round his finger." An 1809 instance offers additional context for the expression by saying "But I can wind him round my finger like a pack-thread."
The early instances of the expression that I found do not explicitly refer to hunting in general or to falconry in particular. Therefore, I think it most likely that the original intended image is of sewing thread being wound around one's finger for temporary storage pending future use.
With regard to the question of when a writer first specified that the little finger was the finger that the threadlike person was being wrapped around, I cannot find an earlier instance than the one Mari-LouA cites from Walter Scott, The Bride of Lammermoor (1819).
Best Answer
"On a wild goose chase" might work.
Cambridge Dictionary gives this definition:
The phrase finder has an article on the history of the phrase. Here's an extract: