I remember seeing something on those electric bulletin boards in Pokemon Black/White that said "nicknames bring better relationships" or something along those lines. I doubt it, but I could be wrong.
Actually, the previous answer was wrong, so I hid it. Nicknames do not increase the happiness of a Pokemon. Nicknaming your Pokemon just shows you care enough to give it a name, a concept that is used in the Nuzlocke challenge.
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