Theoden calls Eowyn sister-daughter. I assume this is because she is, literally, the daughter of his sister.
He also refers to Eomer as his sister-son at one point.
What does Eowyn call Theoden in return?
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As far as I can tell, in the books, Eowyn referred to Theoden by any title only three times. She was very quiet in the Two Towers. She only spoke twice: Once when offering Theoden wine after his recovery, and once when Theoden appointed her as guardian of Edoras as he departed. Neither of those times did she use any address other than his name.
In Return of the King, she only spoke to him a couple of times. On his arrival at Dunharrow:
'Hail, Lord of the Mark!' she cried. 'My heart is glad at your returning.'
He pointed away along the darkening lines of stones towards the Dwimorberg. 'Of the Paths of the Dead?'
'Yes, lord,' said Éowyn. 'And he has passed into the shadows from which none have returned. I could not dissuade him. He is gone.'
Sadly, in the books, Theoden died on the Pelennor Fields before he knew Eowyn was there. She mentioned him once when she awoke in the Houses of Healing:
' I am strangely weary,' she said. 'I must rest a little. But tell me, what of the Lord of the Mark? Alas! Do not tell me that that was a dream for I know that it was not. He is dead as he foresaw.'
This isn't particularly surprising -- in the real world, even the family of British royalty are expected to greet Royalty by "Your Majesty" (and lower nobility by their titles, including "My Lord"). The Queen, however, can call you whatever she damn well pleases.
This is probably a linguistics question. The Rohirric language created by Tolkien has the specific family naming features of Old English that modern english has dropped.
If it's not in the texts, I would speculate Tolkien would have made it 'Éam', which is what I think you're asking. In the books, his role as ruler is primary and calling him 'uncle' in public would probably be disrespectful.
This might be helpful. https://en.wikipedia.org/wiki/Rohirric
As a counterpoint, most cultures of India make even less distinction of familial roles. 'Uncle' means someone of about my father's age, whether they are related or not.