myselfWhat I had never noticed until yesterday was that it's not logical. No surprise, we are talking about English, a language not known for its logicality (not that any languages are, particularly). But I'd never spotted the illogicality in this context before.
If you look again at the forms, they are clearly a combination of a pronoun and the morpheme self, to add the reflexive component. But the pronoun which combines with the reflexive part is either possessive or oblique (a case form other than nominative, which would be I, you, we, he, she etc.).
my - possessive(Her is the same in both forms.)
your - possessive
him - oblique
her - possessive/oblique
it - oblique
our - possessive
your - possessive
them - oblique
There is a pattern: if we assume that her is oblique, then the third person forms are oblique and the others are all possessive. Why should the third person pronouns have a different form from the other persons?
People have suggested various reasons. Maybe it's hypercorrection of me to my, a correction vigorously enforced in education, extended to meself/myself? Not sure - this doesn't explain the other persons. Maybe it's due to use of the third-person vocative (How is His Majesty feeling this morning?)? Doesn't really explain why the third person forms are oblique, though - on this basis they'd be more likely to be possessive.
One thing that we can do is decide which forms came first, and which are later alterations (if any). It seems more likely that the original forms were all oblique: Modern Swedish and Danish use oblique ('objective') forms throughout the paradigm, and Old English did as well. According to the Online Etymology Dictionary (I'm not sure how reliable a source that is; I can't get on the OED from here), myself replaced the older ic me self, where me was a kind of dative pronoun. Ourselves replaced we selfe and us selfum. Themselves is a northern dialect (i.e. from Old Norse) form of heom selfum (again, heom is dative, but so is them so the case didn't change as with ourselves/myself). These changes all seem to have happened around the year 1500. The question is why? The same source suggests that the change from me to my was on analogy with herself, perceived as the possessive, but acknowledges that the equivalent hisself did not become standard.
Of course, hisself and theirselves are both common non-standard dialect forms. Once again, it seems that the dialects preserve the more logical forms (cf. you/yous, or Yorkshire tha/thon/etc.) while the standard is the hodge-podge we've inherited from the various mixed sources. Doesn't it make life way more interesting? And what a good thing we have so many varied dialects to compare to the standard, so we can work out what's going on.