Huma Abedin has been by Hillary Clinton’s side for 20 years, since she was a White House intern assigned to the first lady’s office. There has never been much question that if Mrs. Clinton returned to the White House as president, Ms. Abedin would come too.

Now with Ms. Abedin again part of unwelcome attention, even some Democrats are wondering if Mrs. Clinton might be forced to rethink Ms. Abedin’s role if she is elected president, and whether a cloud would linger over her if Ms. Abedin were to take a senior White House job, as has long been expected.

The Federal Bureau of Investigation said Friday it had found additional emails potentially related to the probe, completed this summer, of Mrs. Clinton’s private server when she was secretary of state. The new emails come from a laptop that investigators believe was used by Ms. Abedin and her estranged husband, former Rep. Anthony Weiner, and were found during an investigation of Mr. Weiner for allegedly exchanging sexually explicit text messages and photos with a 15-year-old girl.

Ms. Abedin now finds herself as a central character in a late-breaking drama that has added an unpredictable factor into what many people in both parties saw as an election heading toward a Clinton victory.

Some Democrats said Sunday they had no doubt Mrs. Clinton would bring Ms. Abedin with her to the White House. Others said her future may depend on what the renewed investigation finds. It is possible the emails are purely personal, were already turned over to investigators, or are duplicates of messages already examined, meaning the long-term damage to Mrs. Clinton—and Ms. Abedin—may be minimal.