I knew that Theoden passed his kingdom to Eowyn but I think Eomer must become king of Rohan - he's the only pure blood royal son.
Éomer did become king following Théoden's death (emphasis mine):
Then a minstrel and loremaster stood up and named all the names of the Lords of the Mark in their order: Eorl the Young; and Brego builder of the Hall; and Aldor brother of Baldor the hapless; and Fréa, and Fréawine, and Goldwine, and Déor, and Gram; and Helm who lay hid in Helm’s Deep when the Mark was overrun; and so ended the nine mounds of the west-side, for in that time the line was broken, and after came the mounds of the east-side: Fréalaf, Helm s sister-son, and Léofa, and Walda, and Folca, and Folcwine, and Fengel, and Thengel, and Théoden the latest. And when Théoden was named Éomer drained the cup. Then Éowyn bade those that served to fill the cups, and all there assembled rose and drank to the new king, crying: 'Hail, Éomer, King of the Mark!'
Return of the King Book VI Chapter 6: "Many Partings"
Though it's worth noting that Éomer is actually Théoden's nephew, not his son. Théoden's son was Théodred, who died off-screen (or off-page) during the events of Fellowship of the Ring
There's only one time I can think of where it would be sensible to say that Éowyn succeeded Théoden as king (well, queen), and that's in Two Towers when both Théoden and Éomer depart for Helm's Deep:
'Behold! I go forth, and it seems like to be my last riding,' said Théoden. 'I have no child. Théodred my son is slain. I name Éomer my sister-son to be my heir. If neither of us return, then choose a new lord as you will. But to some one I must now entrust my people that I leave behind, to rule them in my place. Which of you will stay?'
No man spoke.
'Is there none whom you would name? In whom do my people trust?'
'In the House of Eorl,' answered Háma.
'But Éomer I cannot spare, nor would he stay,' said the king; 'and he is the last of that House.'
'I said not Éomer,' answered Háma. 'And he is not the last. There is Éowyn, daughter of Éomund, his sister. She is fearless and high-hearted. All love her. Let her be as lord to the Eorlingas, while we are gone.'
'It shall be so,' said Théoden. 'Let the heralds announce to the folk that the Lady Éowyn will lead them!'
The Two Towers Book III Chapter 6: "The King of the Golden Hall"
But, as indicated by the above conversation, she was chosen to lead only because Éomer, the King's appointed heir, was off to war. We can imagine that, had Éomer been killed in the Battle of the Pelennor, Éowyn would have ascended to the throne.
Yes, Eomer became king of Rohan after Theoden's death.
Then a minstrel and loremaster stood up and named all the names of the Lords of the Mark in their order: Eorl the Young; and Brego builder of the Hall; and Aldor brother of Baldor the hapless; [...] and Thengel, and Theoden the latest. And when Theoden was named Eomer drained the cup. Then Eowyn bade those that served to fill the cups, and all there assembled rose and drank to the new king, crying: "Hail, Eomer, King of the Mark!"
-- The Return of the King, Book Six, Chapter 6: Many Partings
There are two lines of burial mounds in the hallows where the kings of the Mark are buried. Each line represents lineal descent, from father to son. The first line begins with Eorl. When that line was broken, as there was no living son, the second line began. The second line ended with Theoden because he had no living son. Eomer began the third line.
The way this is presented implies the existence of a rule of inheritance, with eldest living son becoming king, and, if there is no son, then the closest male relative. What would have happened if Eomer had died before he could be crowned is not clear. The people loved Eowyn, but there was no precedent for a woman ruling the Rohirrim.
In the film, Return of the King, on the eve of the Rohirrim’s departure for Gondor, Théoden King says to Éowyn ...“I have left instruction. The people are to follow your rule in my stead. Take up my seat in the Golden Hall. Long may you defend Edoras...if the battle goes ill.“
It appears that Éowyn has been appointed regent until the outcome of the battle is known. If the battle goes well and Théoden returns, he will continue in his position as king. What wasn’t clear to me was what he meant by the battle going ill; does the battle going ill mean that he dies, or that both he and Éomer are killed? Judging by the story, the later was the case, as Éomer survives and becomes king.