What does this aphorism by Emerson mean: "nothing is at last sacred but the integrity of your own mind"?
It is a call to be an Individualist and not a Conformer. It asks that we be Self-Sufficient and not Co-Dependent.
Sometimes quotes are fairly self-explanatory on their own. Sometimes they need a bit of background. This is one in that second category.
Ralph Waldo Emerson was one of the great thinkers of the 19th century and was a leading thinker (if not creator) of a philosophy called Transcendentalism (there's a Wikipedia link below for more information about that movement). As the Wiki article puts it,
"A core belief was in the inherent goodness of both people and nature. Transcendentalists believed that society and its institutions ultimately corrupted the purity of the individual, and had faith that people are at their best when truly "self-reliant" and independent."
This matches up with other quotes of his on the same subject, such as "To be yourself in a world that is constantly trying to make you something else is the greatest accomplishment."
The quote above, therefore, is another in that same vein. For me, it appeals to the reader to shrug off the effects of other people's thinking and to honour your own, to connect with the inherent goodness of the Individual Self and to ignore the masses and the institutions of government and church. It is a call to be an Individualist and not a Conformer. And it asks that we be Self-Sufficient and not Co-Dependent.
As with all quotes, we can take them or leave them, believe them or reject them. The real question is - what do you take from it?