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Edit As Paul Taylor said Florian Cajori favours another root of the root symbol, namely the generation of an upstroke from a row of points. In fact there may have been many sources playing together. Moritz Cantor points out that Alkasadi (or Alkasawi) an Arab living in Spain (died 1477 or 1486) wrote a book which is known under different titles like Lifting the veils of the science of the Gubar (Gubar means "dust" or "calculating with digits") where he not only abbreviated the Arab word for root dschidr by writing its first letter, but wrote it not right of the radicand (in Arabic, meaning in front of the radicand) but above. The jim can be seen in the column Initial in the table Arabic letters usage in Literary Arabic. This could also be a source for our root symbol.

Edit As Paul Taylor said Florian Cajori favours another root of the root symbol, namely the generation of an upstroke from a row of points. In fact there may have been many sources playing together. Moritz Cantor points out that Alkasadi (or Alkasawi) an Arab living in Spain (died 1477 or 1486) wrote a book which is known under different titles like Lifting the veils of the science of the Gubar (Gubar means "dust" or "calculating with digits") where he not only abbreviated the Arab word for root dschidr by writing its first letter, but wrote it not right of the radicand (in Arabic, meaning in front of the radicand) but above. The jim can be seen in the column Initial in the table Arabic letters usage in Literary Arabic. This could also be a source for our root symbol.

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This question has already been completely answered, but here is a bit more on history.

The symbol has its origin in the Latin letter R as an abbreviation of radix (latin: root). It has been used by Leonardo de Pisa (Fibonacci) in his seminal book Liber Abaci (1202) where he treats square roots and cubic roots in chapter 14 and 15 (as well as in his later less well known Liber Quadratorum (1225)).

Fibonacci, like Euclid, did not invent all the mathematics he reports, but took much of it from the Arabian world, mainly from al-Khwarizmi and Omar Khayyam. The Arabian word for "root" had been used by al-Khwarizmi already; his word is rendered radix in translations from the Arabic to Latin by Robert of Chester, John of Seville, and Gerard of Cremona. It appears also in Alexandre de Villedieu's Carmen de Algorismo (1240) and in Sacrobosco's Algorismus (1250). By the way Fibonacci calculates approximations but considers roots as exact numbers, even if they cannot be expressed as integers or fractions.

Following the tradition of medieval writers, Nicolas Chuquet used the uppercase Latin letter R with a small stroke, looking very similar to Px when written close together. Both, R and R$^2$ indicate the square root, R$^3$ indicates the cubic root, R$^4$ the forth root and so on. Regiomontanus (1464), Luca Pacioli (1494), and Estienne de la Roche (1520) adopted this sign.

The hook-like symbol √ that resembles a small r was introduced by Christoff Rudolff in his book "Die Coß" (1525), the first German book on algebra. He used c√ for cubic root and √√ for fourths root. (By the way he introduced also the convention $x^0$ = 1). English, French and Italien writers were slow in adopting that German sign. Even in Germany the symbol "l" for latus (side of the square) was long in use. After Michael Stifel had published the second edition of Rudolff's Coß (1553) the symbol became more and more accepted.

René Descartes (1596 bis 1650) invented (or extended) the bar above the radicand (this word being first used in 1889) in order to indicate what symbols belong to the radicand.

Moritz Cantor: "Vorlesungen über Geschichte der Mathematik", Teubner, Leipzig (1894) http://archive.org/stream/vorlesungenber02cantuoft#page/n5/mode/2up

Florian Cajori: "A History of Mathematics" MacMillan, London (1909) http://www.gutenberg.org/files/31061/31061-pdf.pdf

David Eugene Smith: "History of Mathematics, vol. 2", Dover Publications (1958) http://books.google.de/books/about/History_of_Mathematics.html?id=uTytJGnTf1kC&redir_esc=y

http://jeff560.tripod.com/r.html

http://www-history.mcs.st-and.ac.uk/Biographies/Fibonacci.html