In Japanese culture, children often refer to themselves in the third person (see this question from Japanese Language Stack Exchange, and also the Wikipedia article on Illeism, which is the technical term for referring to yourself in the third person. This JLSE question also gives more of the nuances of using illeism in Japanese.) A Western example of this would be Elmo, the Sesame Street character, although in the West, illeism usually makes someone look big-headed rather than cute, like former WWE superstar The Rock, who always referred to himself in the third person as "The Rock".
In anime, older characters with childish personalities often also use the third person; for example, Bakemonogatari's Nadeko always refers to herself as "Nadeko". (If you see "I" or "me" in the subtitles, it was added by the translators.) Yamada claims to be sixteen if I recall correctly, so she should be too old for this (though the second comment on this JLSE answer claims that even women in their twenties will do this in real life), but she spends most of her time acting like she's much younger than she really is, trying to be cute and make people like her and praise her and overlook her mistakes. She tries to adopt a "poor orphan in search of a loving family" kind of persona and get people like Otou, Yachiyo, and Hiromi to feel sorry for her and join her ideal family. I think her use of her own name is meant to play up her assumed role as the cute, spoiled youngest child of her imaginary family.
You're right that this is different from other times when the subtitles translate a name as "you" or "her"; that usage is more in line with what's described in this Japanese Language Stack Exchange question, where you address someone by name instead of using a second-person or third-person pronoun, as a matter of respect. Going by Yamada's personality, I think it's a childish affectation.