In Batman V Superman: Dawn of Justice, Bruce makes a point to display the Robin costume, complete with Jokers haunting message. We also see a glimpse of the weapon Robin used in combat, and it isn't the familiar quarterstaff we're used to in the comics.
That's par for the course though in Snyder's version of Gotham, a much darker place than we're used to seeing on film. As such it would make sense for Robin to not have a weapon that solely incapacitates foes, but can also go the distance if need be. In the Batman V Superman: Dawn of Justice Tech Manual (via Digital Spy), we can see it had multiple blades and was closer to an ax than merely a staff.
Snyder ended up going with a smashed weapon for the display case, instead of the complete model that they previously had in mind. He most likely would have been using it to defend himself in whatever altercation he was involved in before he died, so it makes more sense to go the damaged route.
As for who the Robin is, still no additional information, but my money has always been on Jason Todd.