This page gives a decent brief accounting of the origin of most of the symbol, but in the cursory perusing I’ve done, I haven’t found any obvious accounting of the little ‘hook’ at the far left of the symbol. I look forward to your feedback, too!
I believe both letters r and R (and several others) have this feature in a lot of scripts.
My WAG is that it is a part of the symbol to eliminate any chance it could be mistaken for a “V” that was mis-struck in printing or mis-written by hand. That hook defines it as a very different symbol than a “V”.
I’m thinking that the first typecutter asked to make such a symbol for printing was a guy who took pride in his work. So he made the downstroke thicker than the crossbar, and he put a serif on the beginning, just as the various numerals have in that typeface.
My go-to source for this sort of thing is Florian Cajori’s A History of Mathematical Notations (two volumes published 1928 and 1929). It’s old so his information may be out of date.
In paragraph 324 of volume 1, he gives an explanation. First he mentions but dismisses an old idea that it’s a modified r for Latin radix.
His preferred theory is that the sign derived from a rather illogical system of dots to indicate roots. In the 15th century one dot before the radicand indicated the square root, two dots the fourth root, three dots the cube root, and four dots the ninth root. In a manuscript from sometime before 1524, the dot had a tail bending upward and to the right so it looked something like a modern musical eighth-note symbol and was followed by a symbol to indicate which root it is (square, cube, etc). Click on the link above to see what it looked like. In a 1525 book by Christoff Rudolff the square root symbol, presumably but not certainly derived from the eight-note-like symbol, looks just like a check mark (it’s shown in figure 59 on page 135 of my print edition, but that page doesn’t appear to be available on Google Books). It’s essentially a modern square root sign but without the vinculum (top bar) and without the little hook at the far left. Other roots were indicated by adding more peaks to left, so that the cube root symbol looked something like a w attached to a checkmark. The vinculum was added by Descartes in 1637 (page 205 of my print edition). He doesn’t say who first added the little hook to the left but figure 79, p. 173, shows an example of the radical sign with left hook and without vinculum from 1592 in a book by Masterson. I suspect Mr Downtown is right that it’s just a stylistic serif-type thing. There is a similar left hook on a cube-root symbol in a book by Robert Recorde from 1557. The page of Recorde’s book that Cajori reproduced doesn’t include any square roots.
On closer inspection, there are two square root symbols on that page of the 1557 book and they both have the left hook.