New Yorkers strolling through Chinatown in downtown Manhattan last Sunday might have noticed an unusual flurry of activity: Jewish men and women, a rabbi in a clerical gown, and a color guard gathering in graveyard tucked away behind a wrought-iron fence. Members of the New York synagogue Shearith Israel, the oldest Jewish congregation in North America, were visiting their historic cemetery at Chatham Square.

In an annual ritual ahead of Memorial Day, they were there for a ceremony that few other synagogues in America could...