Character Analysis
Delio is Antonio's steadfast BFF, and has the distinction of being the only semi-major character to survive the play. Which is really saying something.
Delio's relationship with Antonio parallels Cariola's with the Duchess. Even though he backs Antonio all the way, he's not at ease with Antonio's marriage. Despite being his best friend, Delio's worried about the changes he sees in Antonio, saying "How fearfully / Shows his ambition now…" (2.4.80-81), which echoes Cariola's own discomfort with her mistress's decision to marry Antonio.
Plot-wise, Delio both serves as the only person other than Cariola who initially knows about the secret marriage and, more importantly, as the guy who both opens and closes the play.
Audience Surrogate
The entire first scene of The Duchess of Malfi is structured such that Delio stands in for the audience: he's new to the Malfi court, doesn't know who's who, and needs Antonio to explain it all to him. Together, he and Antonio frame all of our initial perceptions about the court and the people in it.
And given that, like Delio, you the audience don't know anything yet, you have to take what Antonio says more or less at face value. By Delio's final appearance, though, you're probably a lot more skeptical, which makes his closing remarks a lot more subject to doubt.
Right after all of the action with Bosola, the Cardinal and Ferdinand has come to its bloody conclusion, Delio comes onstage with the only surviving child of the Duchess and Antonio to announce his hopes, "To establish this young hopeful gentleman / In's mother's right" (5.5.111-12). For more on this ending (if it failed to totally convince you of the rosiness of this kid's future, you're not a Grinch, you're just sensible), go check out "What's Up With The Ending?"