What does the speaker harbor in "Song of Myself" by Walt Whitman?

By "harbor" I assume you're referring to the thoughts and feelings the poet holds within himself. If that's the case, then the main thought/idea Walt Whitman harbors in "Song of Myself" is the notion of the radically plural individual. For most of the poem, Whitman embarks on an incredibly expansive poetic journey, and he strives to incorporate as much of life (and specifically American life) as possible. As such, he presents a variety of scenes from all different walks of life, both high and low, common and uncommon, and uses these scenes to inform his theory of individual identity. Identity, Whitman claims, is not a static, singular concept, but rather an expansive, lively, continuously changing and at times contradictory idea. As such, he tries to incorporate a vast tapestry of life in order to give voice to his pluralistic (some might even say "democratic") vision of the self and the individual. This theme of the plural self is the heart of Whitman's poem, and so it is likely the most significant idea that he harbors throughout his verse.