Well, I am only an amateur coin collector, but I have done some research on this and can answer it. There are many theories regarding its origin, but the most common among historians and numismatists is that it was inspired by this coin, the Spanish eight reales coin, also known as a dollar, a peso, or a piece of eight. As you can see there are two pillars representing the Pillars of Hercules in the Strait of Gibraltar, these pillars have ribbons around them that say PLUS ULTRA (Latin for "further beyond"). The way these ribbons are arranged, specially the one on the right, looks like a $. You can also see that the mint mark of the coin, minted in Potosi, Bolivia looks as well like a dollar sign. Another theory states that the word for peso in mexico was abbreviated to Ps, with the passing of the years, the s was put on top of the P and made a Symbol similar to the dollar sign, this theory is the most backed up historically.

The symbol spread to the US because Spanish silver coins were trusted by the inhabitants of the colonies and long after the mint was established, this coin continued to circulate for a long time.

Sources:

http://www.bep.gov/resources/faqs.html#wb_element_text__1399408259221_115 https://en.wikipedia.org/wiki/Dollar_sign