Taking tori in symmetric products and "miraculously" proving that the Floer homology is independent of choices always seemed, well, miraculous. Some time ago Max Lipyanski explained to me the origins of this construction from gauge theory on surfaces, a la AtiyahFloer conjecture, which I have then forgotten. What is the origin of Heegard Floer?
I think the crude answer is that there is (or maybe just should be) an extended 4 dimensional TQFT that assigns the Fukaya category of a symmetric product to a surface, and the usual HeegardFloer Lagrangian to a 3 manifold. So, the usual definition of HeegardFloer is the gluing formula for a Heegard splitting, and invariance is no miracle at all. 


From Szabo's delightfully understated response (pdf) to receiving the Veblen prize:
Of course, if one believes that Heegaard Floer homology is somehow the limit of monopole Floer homology as one degenerates the metric in some way that depends on the Heegaard diagram, then the independence of Heegaard Floer homology from the Heegaard diagram would fall out from the metricindependence of monopole Floer homology. Unfortunately, I can't seem to find references that give any sort of precise picture of how Ozsvath and Szabo came to think that this should be the case (though it might have been a baby analogue of the picture in this paper (pdf) by YiJen Lee, written a few years later). It perhaps bears mentioning that Heegaard Floer homology wasn't the first invariant that Ozsvath and Szabo constructed based on thinking about the interaction of the SeibergWitten equations with a Heegaard diagramthese papers, which extract an invariant from the thetadivisor of the Heegaard surface, appear to have been based on thinking about what happens to the SeibergWitten equations when one has a neck Sx[T,T] (S is the Heegaard surface) with the metric on S at t=T itself having long cylinders over the compressing circles for one handlebody, while the metric on S at t=T has long cylinders over the compressing circles for the other handlebody. 


I'm far away from being an expert, but I think the Heegaard Floer homology was invented by Peter Ozsváth and Zoltán Szabó, so I would recommend the following link to you: click me If this Introduction is not enough, you should perhaps read "the original work" (in fact the Heegard Floer homology was developed in a long series of papers): P. S. Ozsváth and Z. Szabó. Holomorphic disks and topological invariants for closed threemanifolds. To appear in Annals of Math., math.SG/0101206. EDIT: Perhaps the Introduction of the book Floer homology, gauge theory, and lowdimensional topology is useful if you are interested in the motivation of Heegard Floer homology. 

