Lodowick makes the really bad move of being the son of Barabas's arch-nemesis, Governor Ferneze. After recovering some of his gold, Barabas's first order of business is to kill Lodowick, on account of having major hate for his dad. Lodowick ends up dying in a duel against his friend, Mathias, when Barabas tricks them into fighting over his daughter.
Lodowick, while more or less a not-terrible-person, definitely has some personality flaws. He's never even seen Abigail before he gets Barabas to fix them up in Act 2, and he's only trying to get with her because (1) he knows that Mathias is interested in her and (2) he thinks she's a babe. Even though marrying somebody you met two seconds ago is par for the course in Renaissance Drama, going after Abigail when he knows Mathias digs her is kind of a major violation of the bro-code, no?
When he meets Barabas, Lodowick introduces himself as the Governor's son, apparently hoping that'll up his chances of making a move on Abigail. Oops. He's really just painting a huge target on his back. It doesn't get better when he calls Mathias a "base-born peasant" (2.3.281). Everything goes downhill from there, and Barabas cultivates the men's enmity until they kill each other in a duel.
Our verdict? Sure, Lodowick doesn't deserve to die. But he's still not someone we want to bring home to mom.