Her emails have been sought by conservative interest groups, she has been accused by Donald Trump of doing State Department favors for the Clinton Foundation, and she has been embarrassed by her husband’s sexual activities.

Huma Abedin, Hillary Clinton’s longest-serving aide, has received an unusual amount of public attention for a behind-the-scenes staffer. Over the years, she has graduated from intern to omnipresent personal assistant to vice chairwoman of Mrs. Clinton’s Democratic presidential campaign and one of her closest...