Does Eowyn Become Queen of Rohan?

Eomer was always next in line to be king, however if he and Theoden had both died in combat, the title would have fallen to Eowyn. Fortunately Eomer did make it and claimed the title as the King of Rohan.

Does Eowyn rule Rohan?

In The Two Towers, Éowyn, a daughter of the House of Eorl and the niece of King Théoden of Rohan, is introduced in Meduseld, the King's hall at Edoras. ... As Aragorn points out, her duty is with her people; she has to shoulder the responsibility of ruling Rohan in Théoden's stead when the war-host of Rohan go to war.

Who is the Queen of Rohan?

White Lady of Rohan, Shieldmaiden of the Rohirrim, Éowyn is a figure who bears the weight of destiny as much as anyone in Tolkien's universe. Yet, she is relegated to a minor role in Peter Jackson's Lord of the Rings trilogy.