The two statements are both grammatically correct, and the difference would be a matter of style, substance, or context. For example, if one was relating about an instance where one was presenting an award, and happened to be the winner of the award he or she was presenting, one might use the latter construction: "I said, 'The best candidate for the position is [opens envelope] ... me?'". On the other hand, in a heated disputation, with someone else who is suggesting a different candidate, one can well imagine a response, "I said I am the best candidate for the position.'"
The second, using "me" has a hint of older usage about it, as it was once preferred to minimize the use of the first person singular. In some older scientific literature this even went so far as to use the third person in writing up research papers, so one usually saw "The authors found that ...", instead of the simpler "We found ...", but this has tended to become less a consideration than formerly, and the more direct first person is preferred.