NEW YORK—Jeffrey Epstein’s longtime lawyer has operated behind the scenes for decades. Now he finds himself in the spotlight over the scandal surrounding the disgraced financier.

Darren Indyke was named co-executor of Mr. Epstein’s will filed earlier this month in a U.S. Virgin Islands court, a document Mr. Epstein signed two days before his apparent suicide. Mr. Indyke and Mr. Epstein had worked together for more than 25 years as Mr. Epstein expanded his business and social ties.

As...