What did Justice Harlan mean when he said "I would rather be right than consistent"?

I assume that he meant that he is willing to rule differently on a matter in the present than he did in the past if he realizes that his opinion was wrong in the past. An example would be ruling against a gay or interracial couple for marrying and now realizing that they should have the right to.

“A foolish consistency is the hobgoblin of little minds.”
—Ralph Waldo Emerson

“When my information changes, I alter my conclusions. What do you do, sir?”
—John Maynard Keynes (attributed)

The two quotes above are basically the same sentiment in different words. Back when Harlan was the Attorney General of Kentucky, he had argued that the federal government had no right to interfere with the practice of slavery. And despite supporting the Union in the US Civil War, he also supported the continuation of slavery. But when he became an associate justice on the US Supreme Court, he regularly sided with civil rights advocates (most famously, he was the sole dissenter in Plessy v. Ferguson). “I would rather be right than consistent” was his response to those who criticized him for having changed his views.

So like @janbb said: he realized he had been wrong, and he changed his behavior accordingly. Harlan deserves quite a lot of praise for this, as it is something that many people find incredibly difficult. Consistency can feel like a safe haven from the discomfort and embarrassment of having to admit that one is wrong, especially when one is surrounded by people who still cling to the same false belief. One has to overcome not only their own biases and mental defense mechanisms, but also the possibility of being rejected by those around them who will not take kindly to the change of opinion. Unfortunately, not everything that’s important is easy.