Found this on the Encyclopedia of Arda;
Quote:
Notes
1 The first syllable of Éomer's name is pronounced approximately like the English word 'ear'.
2 The éo- element of Éomer's name is surely 'horse', but -mer is less certain. It is most likely derived from Old English maer, meaning 'famous, glorious, heroic'.
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Im not sure how correct their sources are, but it maks sense.