Eowyn's relationships are some of the most fraught in The Lord of the Rings. Like all of Tolkien's characters, she Though the romantic relationships she does (or does not) have captivate our attention, her familial connections are also important. Her only surviving family when we first encounter her are, of course, her brother, Eomer, and her uncle, Theoden. The relationship between niece and uncle is particularly interesting.
Though Eowyn is Theoden's niece, it is clear that their relationship is closer than that of uncle and niece. Not only does he call her sister-daughter but also daughter and even dearer than daughter. This is in part due to the fact that Eowyn and Eomer's parents died when they were young, and Theoden raised them. The maternal uncle is a very traditional stand-in for a father. One might think of Beowulf's lord and kinsman Hygelac and Roland's relationship with Charlemagne in The Song of Roland. It is very likely that the Proto Indo-Europeans used a system of kinship terms that did not distinguish between "father" and "maternal uncle." The term sister-daughter used in the text by Theoden instead of niece highlights both the genetic closeness of the relationship as well as the parental-filial nature of it. Eowyn's love for her uncle is also clear. When she sees him under the influence of Wormtongue, she pities him. Her love for her uncle is clearest during the Battle of the Pelennor Fields: "The knights of [Theoden's] house lay slain about him, or else mastered by the madness of their steeds were borne far away. Yet one stood there still: Dernhelm [Eowyn's alias] the young, faithful beyond fear; and he wept, for he had loved his lord as a father."
Soon after this sentence comes one of the most tragic moments in The Lord of the Rings is when Theoden lies dying: "'Bid Eowyn farewell!' And so he [Theoden] died, and knew not that Eowyn lay near him." In some ways it is sadder than tragic. After all, Greek tragedies have their moments of recognition. The audience might fear that Electra will not recognize Orestes, but she always will. Sometimes this moment comes to late, as in Oedipus Rex, but that play hinges on the fact that wife and husband will see that they are mother and son. In Greek tragedies, the moment of recognition is often the climax (as when Clytemnestra recognizes her son just before he kills her). This revelation and recognition is integral to many happier stories as well. In fairy tales, for example, the true identities of the characters are always revealed at the end: the stable boy is recognized as the king's son, the hero realizes that the magical animal that has been helping him is his mother under a spell, the old crone turns into a beautiful maid. Mary Magdalene, Thomas, and the disciples on the road to Emmaus recognize the risen Christ. These moments are necessary to complete the arc of the narration and to give closure to both the audience and the characters. This is not always possible in the real world, however. The truth is not always revealed at the ideal time. Peter Jackson could not do it. Instead he gives to Theoden the opportunity to realize Eowyn's sacrificial love and battle prowess and to Eowyn the validation of her father-figure. (As an aside, this might be the reason that Eowyn's relationship with Faramir is only present in the extended version of the movies.)
This inability to realize that Dernhelm was Eowyn must mean that in between the time when Theoden's horse falls on him and he speaks to Merry, he was unconscious or at least unable to hear and see Eowyn reveal herself. This moment, however, points to a more profound inability on the part of Theoden to recognize his beloved niece. This can also be seen when Hama is advising Theoden about who should be regent while Theoden is away at war. He has just named Eomer as his heir, but he "cannot spare" Eomer, "nor would [Eomer] stay." None of Theoden's men volunteer to act as regent instead of going off to war. He then asks whom the people trust. Hama tells him that the trust "In the house of Eorl." Theoden assumes this to refer to Eomer, but Hama corrects his lord: "'I said not Eomer,' answered Hama. 'And he is not the last. There is Eowyn, daughter of Eomund, his sister. She is fearless and high-hearted. All love her. Let her be as lord to the Eorlingas, while we are gone.'" Theoden readily accepts this correction. This shows that he agrees with the assessment of Eowyn's ability and aptness to lead his people. Why did he not think of her immediately, then? The most basic answer is that he was not able to recognize her. In many ways, her place at court resembles that of Beowulf's Wealhtheow, the wife of Hrothgar. Both are beautiful women with a noble bearing, closely connected with an aging and at times helpless monarch (even though Theoden comes out of his impotence when he is free of the influence of Wormtongue). They are the gracious ladies who brink drink to the lords. Both are greatly beloved of the king. Theoden's blindness to the true potential of Eowyn seems to come from an inability to think of her outside of this context. Even after acknowledging that she is as much a member of the house of Eorl as her brother, he does not consider that, like her brother, she might refuse to remain behind. Theoden "cannot spare" Eomer, but as it turns out, Eowyn is the one he will need in battle. She cannot save his life, but she does save him from the ignobility of being eaten, mutilated, and without a faithful retainer by his side.
This is not to say that he does not understand her at all. When Aragorn has gone to the Paths of the Dead, Theoden perceives her concern and sorrow. Her love for and worry about Aragorn make sense for a woman of her status at Theoden's court. Perhaps he notices it because he expected it.
It seems then, that you can love someone dearly without understanding them completely.