Abigail is Barabas's 14-year-old daughter, and may very well have been the only character in this play you actually liked. Early on, Barabas claims that the only things in the world that he cares about are Abigail and money, but if you've read the whole play you know how that turns out. (Spoiler alert: not well.)
Abigail loves her father dearly, and when she finds out that Ferneze has taken all of Barabas's wealth and possessions she's willing to do whatever to help her father. She pretend-converts to a nun so that she can enter Barabas's house-turned-nunnery to recover his hidden stash of gold, and later goes along with the instructions he gives her in the course of his plot against Mathias and Lodowick.
So far? Score one for daughterly duty.
When she finds out that Barabas has tricked the two guys into killing each other, though, Abigail literally has a Come To Jesus Moment. She's not happy about Lodowick's murder, but she's really unhappy about Mathias. Why?
This is the moment where she loses faith not only in her father, but in worldly society in general, saying "But I perceive there is no love on earth, / Pity in Jews, nor piety in Turks" (3.3.47-8). Since all of these groups of people have proved to be equally terrible, Abigail turns to Christianity.(Guess she's never heard of the Inquisition.)
Something to note, though: she's not turning towards the Christians, exactly, since she's not going to be living as a normal citizen with people like Katherine and Ferneze. Instead, she wants to be a nun.She's effectively turning away from the "loveless earth," telling Friar Jacomo that she "was chained to the follies of the world" (3.3.60) but now truly wants to become a nun.
In the big picture, Abigail does two main things:
First, she shows up hypocrisy of basically every other person on Malta. Abigail calls it like she sees it. She's furious with Ferneze for screwing over her father, but she also recognizes that Barabas is seriously crossing a line when he arranges thedeaths of Mathias and Lodowick. So, she does the only thing that she can do: become a nun. The only non-bigoted major character of this play has to leave the earthly world of Turks, Jews and Christiansto maintain hermoral integrity.
Second, she tells us a lot about Barabas. Barabas introduces himself by stating that he cares about two things: money and Abigail. But watch all the moments where Barabas conflates the two: when she manages to sneak his hidden stash of gold out of the nunnery to him, he giddily rambles "Oh my girl, / my gold, my fortune, my felicity/…Oh girl, oh gold, oh beauty, oh my bliss" (2.2. 48-55).
And later, when craftily arranging with Lodowick to hook him up with Abigail, Barabas repeatedly refers to Abigail, his virgin daughter, as an "unfoiled diamond." In other words, Abigail is going to inherit his wealth by she's also a part of his wealth. To get out from this hopelessly money-obsessed system, she has to cut herself off both from her father and society at large.
Check out the moment Barabas turns his back on Abigail. When he finds out that Abigail is going to convert, he immediately curses her and then plots her murder. Since he quickly and callously decide to kill the one person he's ever professed (there's that word again) to care about, we know that he's a Really Bad Dude.
No more Mr. Nice Guy. Or even Mr. Not-So-Nice Guy But With Understandable Motivations and Feelings. Barabas has become a monster—and Abigail helps us see it.