Well in the first place it can add an absolute element of uncertainty to the workings of our minds. On the smallest level our thoughts cannot be reduced to simple statement and pattern. This, combined with suitably twisted thinking, can be interpreted as allowing an escape for those among us who are material determinists. Even if we can see and analyze all of our pathways of thought on a material level there is still an element of the unknown and unknowable, an area where cause and effect become a little hazy.
You could use it to argue for idealism, in the finest traditions of Berkely. With quantum theory the observer or act of observing has a discernable effect on the results, so it could be argued that our perceiving the world at any given time in some small way solidifies it. Maybe it would be more accurate to say our observation of one thing occuring collapses the possibility of anything else existing in the same space/time point, even if these interactions are happening on the subatomic level. You seeing it makes it so (not creating so much as holding in place).
Epistemologically Im not sure what it means. Does the fact that some things cant be known invalidate those which presumably can? There is also the idea that there are things we CANT know to contend with. The notion that there is information out there that no matter what we do will be completely beyond us. There seems to be a general unspoken assumption that if we can conceive of a kind of data, we will be able somehow to get it; that if there definitely is an answer that we should be able to find it. Its also the first time in science we have been faced with an either/or descision in the quest for knowledge. Admittedly this is on a tiny and possibly entirely inconsequential scale, but the fact remains that before we never had to choose between knowing one thing and knowing another (of the things we can know that is), we could always know both with sufficient time, money and resources. We never had to choose between, say, knowing an elements atomic weight and knowing its atomic number. Its a little thing but it sets an epistemological precedent I think.
What we do know can also now be looked at as "What we know out of what we can", for if there is one value that cant be known, why not more? Its not the first time we have encountered the concept, especially if you have studied religion, but it is the first time it has been imposed on us by necessities of physical law. This could end up with us going in circles, asking ourselves if we can know whether we can know what we can/cant know....we could end up trying to look down our own metaphorical earhole, so to speak.
Fortunately Im an agnostic so the idea of the unknown doesnt bother me so much :)
I apologize if my knowledge of physics is flawed, I havent looked at quantum theory for some time. Please do correct me if I am wrong.
Hope that was of some help, or made even the slightest bit of sense.