[Anunia]

I have the same problem since I read for the first time LoTR... Now I looked in the Encyclopedia of Arda... Here's what it says about this...

ÉOMER
Pronunciation: earme'r

Notes
1 The first syllable of Éomer's name is pronounced approximately like the English word 'ear'.
1 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'.


ÉOWYN
Pronunciation: Approximately 'earw'in'

Notes
1 The diphthong éo is effectively extinct in English, and pronunciation is doubly difficult since Tolkien's example, the name Theobald, is also rare. The vowel-sound of the word 'ear' comes close to the correct pronunciation.

Theobald, incidentally, is a real Old English name, of the kind that might have been used by the Rohirrim, though they would likely have rendered it Théodbold. It combines the elements théod 'people' and bald 'bold'.