Rather than any of these, the best phrase would actually be "a personal matter," because it states that you are dealing with something personal, but it does not imply anything about what you are dealing with; it is non-specific, and that's a good thing here.
"Personal issue" implies that it is a more troublesome (possibly even emotional or mental) matter to deal with, and you don't want your boss to think you have "issues," which might mean you are having trouble dealing with life in general.
"Private issue" would be somewhat misused here, because it is more typically used with regard to a product (often stocks or bonds) that are "issued" in a non-public way. ("Private matter," on the other hand, might possibly be used if you want to emphasize that the matter is not merely personal, but is in fact something you do not want to reveal or discuss.)
"Personal affair" carries a faint implication of either a romantic liaison, or at least something frivolous, and you don't want your boss to think anything of that kind.
"Personal business" might be all right, but you may not want to use it because the word "business" in this context carries the subtle implication that you have another job or your own company of some kind, and you certainly don't want your boss to think that.