Elanor has it right:
Quote:
Eowyn is of course a girl, and therefore Theoden's neice. Eomund was Theoden's brother-in-law: he was married to Theoden's sister Theodwyn, and Eomer and Eowyn were their children.
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That is why Theoden calls Eomer "Sister-Son": in other words, Eomer the son of my sister Theodwyn.