Nothing concrete, of course. It's all just for fun.
Jon Negroni, of The Pixar Theory, which is a real book you can buy, says that Jessie's original owner was Andy's mom based on the fact that Andy's old cowboy hat seems to be the same kind of hat as Jessie's hat. The theory says that Andy's mom, who would eventually abandon Jessie, also had a Jessie hat and eventually gave the hat to Andy.
Of course, there are some problems with the theory. The details of the hat don't match up aside from the brim decoration and the base color. It also doesn't make much sense that Andy's mom would throw Jessie away, but not the hat. It also doesn't make sense that she'd keep Woody, which is seen as a part of a set with Jessie, but throw only Jessie away. Even if Andy's hat was indeed Jessie's hat, seeing how Woody's Roundup was supposed to have been a very popular franchise, it wouldn't be too farfetched to say that Andy's mom was one of many people who had the hat as a child, and she doesn't necessarily have to be that person who abandoned Jessie.
Oh well. But it's a fun exercise in weaving a story out of one of the most beloved series in modern cinema. And I wouldn't put it past Pixar employees to think up an intricate backstory that we never get to see.
What do you guys think?