Historians aren't entirely sure! He doesn't really seem like a very good choice, and there aren't very good records of why the decision was made when it was.
The decision was made all the way back in 1869 by the US Treasury, but their own records do not provide any clear rationale for choosing Jackson. The decision was made in secret, by unnamed bureaucrats, and approved by the Secretary of the Treasury. It may have had something to do with his populist image; he grew up a poor orphan and yet went on to become not only a wealthy lawyer but President of the United States. With his true rags-to-riches story, a lot of people saw Andrew Jackson as personifying the American Dream.
Still, it's kind of baffling that he was chosen, because Jackson was strongly opposed to the idea of central banking and paper money! Combine that with the history of grave crimes committed by Jackson (particularly against Native Americans), and he seems a more embarrassing choice all the time.