Why is a bit of a tough question, but I can at least give you an idea of when it started.

The oldest-known surviving text about Robin Hood, from the early 15th century, mentions his wearing a hat (specifically, "in hat and hood" is the description). At the time, hoods were fairly common headgear in England, regardless of social status; however, hats denoted a higher status, so this line would've immediately marked Robin Hood as someone with a bit more wealth or social class to a contemporary reader. (Margaret Scott's Medieval Costume and Clothing)

Why this hat specifically though? Well, that may actually be a product of medieval thrift, interestingly. The first such image of Robin Hood comes from "A Lytell Geste of Robyn Hude," which was a very popular telling of the story (and one of the oldest complete narratives) printed by Jan van Doesbroch in Antwerp in the early 16th century. The illustration of Robin Hood was similar to the modern iconography, and anyone familiar with the Errol Flynn or Disney versions would recognize it. But it wasn't originally drawn for Lytell Geste; it was actually an illustration of the Knights Yeoman from Canterbury Tales, which the same printer had printed just earlier. It's likely it was reused to save money on having a new press designed and created.

If you read the description of the knight's yeoman from Chaucer, you can definitely see where Robin Hood could've descended from.

https://www.owleyes.org/text/canterbury-tales/read/the-yeoman#root-218778-1

It's well known that certain scenes in Disney's Robin Hood re-used animation from the earlier the Jungle Book - so maybe that was to save money for a struggling studio, but I like to think it was honoring a tradition in portraying Robin Hood.

(Sourced from "Images of Robin Hood" by Joshua Calhoun and Lois Potter)

(Many interesting answers already, but I'll add a couple of things even though it means repeating what other people have said already)

According to John Marshall (2008/2020), the association of Robin Hood with the bycocket was not mentioned in writing in the original tales and ballads, even though the hat was common from the 14th to 15th century and worn by men and women. However, it may go back to one of his first known pictorial appearancs, a woodcut in the Gest of Robyn Hode, attributed to Antwerp printer Jan van Doesborch, printed sometimes between 1510 and 1515. The image was not original: it recycled one used in the 1492 print of Chaucer's knight yeoman in the Boke of the tales of Canterburies, by Richard Pynson (as mentioned by u/Ad_Homonym_).

If the hat in these images does not match our idea of a Robin Hood hat, it is because (again according to Marshall), Robin is shown wearing it backwards, like people today wear baseball caps, except that this was done to prevent losing the hat when it was worn when hunting or working in the open. Not everyone is convinced by Marshall's interpretation (see the review of Driver, 2010), but the illuminated manuscript Luttrell Psalter (circa 1320-1340) does indeed show characters wearing their bycocket backward: one is a monkey (!) driving a cart (go to folio 19, bottom center), the other is a peasant behind a plough (folio 20, bottom left).

Does this mean that this hat was the ur-hat? Not really.

The front page of the 1550 edition of A mery geste of Robyn Hoode shows Robin with a feathered hat, probably made of fur, and not a bycocket. The guy on the right is Little John, wearing an armour.

The fact is one can browse through centuries of Robin Hood images without seeing The Hat. Let's have a look at Joseph Riston's collection of Robin Hood stories and ballads of 1795, a popular and often reprinted book that kickstarted the Robin Hood revival of the 19th century and an avalanche of Robin Hood books and plays. Riston's book contains numerous illustrations by Thomas Bewick, with costumes that are more from the 17-18th century than from the Middle Ages (Bewick was more into animals). The hat worn by Robin is a feathered one and nothing special.

Up to the 1850s, depictions of Robin Hood hats are similarly disappointing:

• 1810, Robin Hood's garland

• 1810, The Life, Death, and Adventures of Robin Hood and Little John

• 1820, Famous Exploits of Robin Hood; including an account of his birth, education, and death

• 1841, Robin Hood and his merry foresters (illustrated by John Gilbert)

• 1846, The Life and Ballads of Robin Hood, the Renowned Sherwood Forester. This one at least makes it more or less medieval.

In 1850, John Matthew Gulch published a revised edition of Riston's collection, with illustrations by F.W. Fairholt, including this one and this one. We're getting there! Not only the costume is medieval-looking but Robin Hood gets a special feathered cap, that looks a little like the one worn by Douglas Fairbanks in the 1922 Robin Hood movie. Not pointy like a proper bycocket, but at least recognizable.

Robin still gets all sorts of caps in the latter half of the 19th century that slowly evolve into the final one (eg 1855, The Life and Exploits of Robin Hood) but the book that really started to define the "Robin Hood style" was the successful The Merry Adventures of Robin Hood by American illustrator Howard Pyle (1883). As we can see here, for instance, the hat is still not a bycocket, and Robin and his companions wear different types of hats. Pyle's illustrations, of course, are gorgeous, and the book became a classic. British illustrator Patten Wilson gave Robin Hood a similar hat in Ebutts's Hero-Myths and Legends of the British Race (1910).

For Robin Hood to get a proper bycocket with a pointy end, we need to wait until 1912, when American illustrator Louis Rhead published his Bold Robin Hood and his outlaw band. Now, this is the final iteration of the Robin Hood hat and Robin and his companions wear it throughout the book. It is similar to the Errol Flynn version of 1938, which, along with the Disney movie, definitely popularized the style as the "true" one.

Also in 1912, a Robin Hood movie was released, where characters wore bycocket hats including ridiculously oversized ones. This may indicate that there were some popular illustrations with bycocket hats published before 1912. But, in any case, the association of Robin Hood with a special hat can be dated from the mid-1800s (the feathered cap) and the bycocket emerged in the late 1800s - early 1900s.

So, after something of a false start in the 15th century, it took 400 years for Robin Hood to get a proper hat.

Sources

• Amphlett, Hilda. Hats: A History of Fashion in Headwear. Courier Corporation, 2012. https://books.google.fr/books?id=Gx7CAgAAQBAJ&pg=PA34.
• Driver, Martha W. Review of Review of Images of Robin Hood: Medieval to Modern; Robin Hood: The Early Poems, 1465–1560: Texts, Contexts, and Ideology, by Lois Potter and Joshua Calhoun, Thomas H. Ohlgren, and Lister M. Matheson. The Journal of English and Germanic Philology 109, no. 3 (2010): 416–20. https://doi.org/10.5406/jenglgermphil.109.3.0416.
• Ebbutt, M. L. Hero-Myths And Legends Of The British Race. London: George G. Harrap & Company Ltd, 1918. http://archive.org/details/in.ernet.dli.2015.45570.
• Marshall, John. ‘Picturing Robin Hood in Early Print and Performance: 1500–1590: From: Images of Robin Hood: Medieval to Modern (2008)’. In Early English Performance: Medieval Plays and Robin Hood Games, edited by John Marshall and Philip Butterworth. Routledge, 2019. https://doi.org/10.4324/9780429427787.
• Pyle, Howard. The Merry Adventures of Robin Hood : Of Great Renown in Nottinghamshire. New York: Scribner, 1883. http://archive.org/details/merryadventureso00pyle2.
• Rhead, Louis. Bold Robin Hood and His Outlaw Band : Their Famous Exploits in Sherwood Forest. New York ; London : Harper & Brothers, 1912. http://archive.org/details/boldrobinhoodhis00rhea.
• Riston, Joseph. Robin Hood: A Collection of All the Ancient Poems, Songs and Ballads, Now Extant, Relative to That Celebrated English Outlaw; to Which Are Prefixed Historical Anecdotes of His Life, Vol. I and Vol. II (Both Volumes Bounded in One). London: T. Egerton, Whitehall and J. Johnson, 1795. https://archive.org/details/dli.granth.72501/page/12/mode/2up.