I think that Frodo wants to believe that there is some possibility for redemption in Gollum, because he wants to believe that there is some possibility for redemption in himself. By the time he reaches Mordor, Frodo can feel the pull of the Ring grow stronger and stronger. The hopelessness of his situation becomes more and more apparent, so Frodo clings on to the last shreds of hope that he can find. If he finds that hope in Gollum, then so be it.
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But Melkor also was there, and he came to the house of Fëanor, and there he slew Finwë King of the Noldor before his doors, and spilled the first blood in the Blessed Realm; for Finwë alone had not fled from the horror of the Dark.
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