Warning: The following contains SPOILERS for Batwoman season 1, episode 17, "A Narrow Escape ."
Batwoman season 1, episode 17, "A Narrow Escape" confirmed that the Arrowverse's version of Lucius Fox was killed by one of Batman's enemies, Tommy Elliot aka Hush. While Elliot didn't pull the trigger on Fox himself, he employed the crooked cop and member of Crows Security who was exposed as Fox's killer.
The murder of Lucius Fox has been one of the core mysteries surrounding Batwoman season 1. Originally it was believed that a career criminal named Reggie Harris had killed Fox during a botched convenience store robbery, but it was revealed during episode 6, "I'll Be Judge, I'll Be Jury", that the lead detective, prosecuting attorney, and judge on the case were all crooked. This resulted in new trials for a number of convicted criminals, including Harris, much to the dismay of Lucius' son and Batwoman's tech expert, Luke Fox.
However, it became apparent that there was some kind of conspiracy surrounding Lucius' death after Luke confronted Reggie Harris the day after he was released from prison in episode 16, "Through The Looking Glass". A sniper killed Harris just after he told Luke that he and Lucius had talked about Luke's graduation while waiting in line together at the store. It was later revealed that the owner of the convenience store where Lucius died was also shot by a sniper that day and that they had lost the security footage of the night in question shortly after receiving a $50,000 payment from Crows Security. A sniper also tried to kill Sophie Moore and Jacob Kane, who were investigating the unreasonably large pay-off for "damages to a business" from inside the Crows organization. As it turns out, all of this stemmed from Elliot hiring someone to obtain Lucius Fox's personal journals.
"A Narrow Escape" saw Gotham City being rocked by a series of bombings believed to be the work of a terrorist known as The Detonator, whose work hadn't been active in seven years. These bombings were ultimately revealed to be a part of the conspiracy to cover up Lucius Fox's death, with the bombings targeting key locations and people who might have exposed the conspiracy, such as Reggie Harris' defense attorney, Jacob Kane, and Luke Fox. The efforts of Kate Kane, Sophie Moore, Julia Pennyworth, and Luke Fox combined revealed the head of the conspiracy and Lucius Fox's killer as Miguel Robles; the current head of Crows Security's homicide investigations division and a former GCPD cop who had worked on The Detonator case as a rookie.
When Batwoman caught Robles, he confessed that he had only gone after Fox for a payday, having been offered $10,000 by Tommy Elliot to rough-up Lucius Fox and acquire his journals. The grand irony is that Robles had never meant to kill Lucius Fox and that his gun went off accidentally after Lucius charged him, enraged after Robles threatened to go after Fox's family if he didn't give Robles the journals. Everything that followed was born of Robles' increasingly desperate attempts to hide his corruption. Precisely what was in Fox's journals that interested Tommy Elliot is unknown. Yet with Elliot having put in an appearance in "A Narrow Escape" and having been confirmed to be in Arkham Asylum, it seems likely that he'll soon be appearing in his villainous identity of Hush on Batwoman sometime soon.