My own position is that Rand's mature thought is not in any significant way influenced by Nietzsche. Not only is her ideal of man very different from Nietzsche's, but so is her individualism. Nietzsche believed that great men are forged in the crucible of brutal myths (e.g., the myth of eternal recurrence) and harsh adversity ("what doesn't kill us outright makes us stronger"). Rand believed that rational thinking, when zealously followed, lead to the complete integration of emotion and thought, which formed for her the bedrock of human greatness. Nietzsche identified freedom with the battle for freedom; Rand with the attainment of free market individualism. Nietzsche believed in the inevitability of conflict; Rand argued that no conflict was even possible between rational men. Nietzsche equated skepticism with honesty and regarded convictions with immense suspicion; Rand equated even the mildest forms of skepticism with a concerted attempt to undermine man's cognitive faculty. Nietzsche contended that nearly everything men believed were lies, but that some of these lies were psychologically or socially useful; Rand insisted that all lies are bad and that no good can come from believing something that is not true.
This brief sketch should demonstrate that, regardless of the few superficial resemblances between Nietzsche's philosophy and Rand's, they are nevertheless very different philosophers. For those that may still have a few doubts on this score, consider the following. Rand evinces no real understanding or insight into Nietzsche's thought, despite having read at least three of his books. She says of Nietzsche that he advocated sacrificing "others to oneself." Of course, Nietzsche's view on egoism, altruism, and self-sacrifice are actually far more complicated and ambiguous. Consider the following quote from Nietzsche's Anti-Christ:
When the exceptional human being treats the mediocre more tenderly than himself and his peers, this is not mere politeness of the heart—it is simply his duty
Rand's other major complaint against Nietzsche is that he disowned reason in his book The Birth of Tragedy. If Rand had bothered to read the preface of that book, she might have realized that the work is not representative of Nietzsche's mature thought, that he turned against the sort of romantic Wagnerism expressed in its pages and became much more sympathetic to science and even "reason."
Now if Rand didn't even understand Nietzsche, how on earth could she have been influenced by him? Perhaps a case could be made that she was influenced by what she read into Nietzsche; but she was hardly influenced by what Nietzsche actually wrote and meant.