I have no clue how I did it. I was setting someone's contract as a euro-stash to make room for one of the new players being signed and hardship was just on the list. I haven't been able to do it with any other players or teams on my roster. I'm guessing there has to be a certain number of players on the team with an injury? Or maybe it's related to the duration of the injury?
Strangely, the player the hardship option popped up for was Dario Saric, who is actually one of the injured guys on the roster. But like I said, I haven't been able to duplicate that with any other player healthy or injured.