First of all, we must recognize that this is a matter of opinion. There is no objective way to determine whether a president is a “fraud.”
Second, we must also recognize that it is hard to know how Hayes would have lived up to this reputation. The reputation did not have anything to do with what Hayes did, but rather with how he had gotten elected. Hayes was elected in a vote that was hotly disputed. It was not clear which delegations from various Southern states were legitimate and so no winner was announced until 1877. Then, Hayes was named president in the “Compromise of 1877.” Some newspapers referred to Hayes as “His Fraudulency” because they felt that he had not really been elected.
Once in office, it is not clear what Hayes could have done to prove (or disprove) that he had been elected through fraud. It is not as if bad performance once in office would prove that he was elected fraudulently. After all, there are plenty of presidents who have been legitimately elected and have, in some people’s eyes at least, governed very poorly.
Hayes clearly did not govern as if he were trying to pay off debts from his election. He was aggressively honest while in office, in contrast to how things had been in the Grant administration. He spent a lot of effort on trying to get federal officials to be appointed on merit, not on political connections. In other words, Hayes seemed to be trying to govern honestly.
For this reason, I would say that Hayes did not live up to his nickname, but that is simply an opinion that cannot be objectively proven.