[LePetitChoux]

I think Grima would have wanted to marry the daughter, because she was the sucessor to the throne, and that is probably why he wanted Éowyn in the first place (it's not like he'd had enough intelligent conversations with her to like her personality). Also, Éowyn would probably feel even more excluded, because the princess would have had all the attention, and perhaps Éowyn would be a "Cinderella" figure- a member of the family who was there, but seemingly just there because she was family. Theoden would have no need for her. He wouldn't bother being a father figure, because he had an actual daughter to be a father to. She wouldn't be the woman of the house, for the princess would inadvertently steal that role too.
Now, Éomer is a different matter altogether. He probably would have more respect from Theoden's court because he was a successful fighter, and actually went out and did stuff (killing enemies to earn glory etc.).
I think Tolkien liked Éowyn too much to let that happen to her, and he didn't want the main feeling for her to be pity, as it would have been had the horrors above taken place, so he decided to laisser tomber to Rohirric Princess.