If I had to guess, I'd say the idiom's popularity persists because the smallest finger in your hand is the most vulnerable looking one, the one we tend to use the least and the one we pay least attention to. Consequently, a woman who wraps somebody (the "victim" is usually a man) around her little finger, suggests that despite her seemingly meek nature and modest physical strength, she is powerful enough to bend and manipulate a man at her will.
Dictionary.com provides this information on the idiom
Twist around one's finger
Also, turn or wind or wrap around one's finger. Exert complete control
over someone, do as one likes with someone, as in Alison could twist
just about every man around her finger.
This hyperbolic term dates from the mid-1800s.
The date stamp, mid-1800s, seems to contradict the belief that this idiom originated in medieval England. Try as I might I could not find confirmation of the latter, although Wikipedia supports the falconry parentage it fails to provide any specific dates.
Trivia: Interestingly, the little finger in medieval times was known as the auricularis, or 'ear-finger' from its use for cleaning the ears.
The Senses in Late Medieval England
I checked with Google's Ngram Viewer to see if there were any recorded instances of the idiom before the mid-18th century. It appears there isn't, the earliest one I found is dated 1893 in a book entitled The Success of Patrick Desmond by Maurice Francis Egan
Ngram results with the expressions: twisted around her finger, wrapped around her finger, wrapped around her little finger, and wound around her little finger
As I was finding it difficult to believe that such a well-known phrase did not have any recorded evidence prior to 1893 I changed my criteria. This time I searched for the terms: around her finger, round her little finger, around her little finger, and round her finger and limited the time scale between 1750 and 1920.
1826 Head-pieces and tail-pieces, by a travelling artist by Leitch Ritchie
She was the light of her father's eye, and the pride of his heart, and
so complete was her dominion over his affection, that, in the common
phrase, she could have turned' the old man round her finger.
1819 The bride of Lammermoor by Sir Walter Scott
I thank ye, Craigie, and pledge you—I see nothing against it but the
father or the girl taking a tantrum, and I am told the mother can wind
them both round her little finger.
1780 The Town and Country Magazine, Or Universal Repository of Knowledge, Instruction and Entertainment
Most of the limbs of law do everything in a jiffy; but ask what they mean, and they would be as much puzzled, [...] if such gibberish were confined to hackney clerks of twelve shillings a week, we should not notice it (...) but the misfortune is, by degrees it has found its way into more polite assemblies, and a lady of taste was heard to say the other evening at the Pantheon, that she could turn Sir William B_____ round her finger in a jiffy.
If people of sense or common understanding, would reflect one moment
on the folly of using words and phrases they could not explain, they
would certainly explode them, and shun those who used them, as being afflicted with a verbal contagion, ...
Conclusion and Summary
Not one excerpt I sifted through mentioned falconry in any form or shape e.g. "Like a falconer she would wind him round her finger" Which sounds awfully suspicious, such a strong and powerful metaphor must have been written down at least once! And yet I didn't find any evidence to support the theory that the idiom derives from a falconer's practice.
If we revisit the Wikipedia page on falconry we learn
In the UK and parts of Europe, falconry probably reached its zenith in
the 17th century, but soon faded, particularly in the late 18th and
19th centuries, as firearms became the tool of choice for hunting
(this likely took place throughout Europe and Asia in differing
degrees). Falconry in the UK had a resurgence in the late 19th, early
20th century during which time a number of falconry books were
published. [emphasis mine]
I have found evidence that the idiom existed before the mid-19th century (1790) but none dating back to the 17th century or earlier. If this idiom really had origins in Medieval England, as a few have proposed, there would be some trace in English 15th or 16th century literature. I didn't find any.
I put forward that the metaphor began life as "to wind/turn/twist/ someone round one's finger" the adjective, little, was added later as a flourishing touch.
Furthermore, I suggest a far more logical and simple analogy as to how this idiom arose. When a young lady sees a man that fascinates or intrigues her, one of the tell-tale signs that betrays her interest is the twirling of a lock of hair usually around her index finger. The little finger's position and its limited dexterity makes this action highly unlikely.
Image from wikiHow
Several modern online guides confirm this subconscious/instinctive behaviour:
However, fiddling with hair is also another non-verbal body language
that can signify different things in different situations. For
example, hair twirling can signify interest and desire. Women in
particular are noted for hair twirling when talking to someone that
they’re attracted to
Body Language Expert
Gentle movements, such as mindlessly twirling a strand around her
finger or running her fingers through it slowly are signs that she's
Wrapping the hair around a finger is a gesture that mimics the
innocence of childhood and is often used as a flirtation device.
"Twisting" in 19th century
1819-1820 The Atheneum, or, Spirit of the English Magazines
1813 Memoirs of Prince Alexy Haimatoff