Elliott Management Corp., known for its brawls with CEOs and developing-world governments, is softening its image.

As the shareholder-activism landscape evolves and the influence of big index-fund managers like BlackRock Inc. and Vanguard Group grows, Elliott has been cultivating better relations with those investors and the companies it targets.

To spearhead the outreach to those investors, Elliott tapped its 30-year-old former research analyst, Christine O’Brien, whose job it is to market the firm as a force for sound corporate governance.

Ms. O’Brien’s mandate is to highlight “the corporate-governance work Elliott has quietly been doing all along,” she said in an interview.

Some activist-defense advisers and institutional investors, none of whom would talk on the record for fear of hurting their relationships with Elliott, say Ms. O’Brien has helped humanize the firm and shift its focus away from winning support in any one fight and toward building long-term relationships.