In Othello, what does Iago convince Roderigo to do?
Roderigo is a fellow who is easily led by Iago. So, he could, potentially, convince him to do any number of things. Roderigo pretty much does whatever Iago says.
In Act I, scene i, Iago convinces Roderigo to be the man standing in the light under Brabantio's window, when they rouse him to tell him that his daughter has married Othello. The talk is lewd and bigoted, lots of it supplied by Iago, who hides out of sight. So, here, Iago has convinced Roderigo to "take the fall" for the bawdy conversation under Brabantio's window. This also saves Iago from any implication in condemning Othello in any way, which is key, if his plan of pretending to be loyal to Othello is to go forward.
Later in Act I, scene iii, he convinces Roderigo to follow Othello and Desdemona to the wars, not to give up on his love for Desdemona. But, curiously, he also convinces him to do one more thing. "Put money in thy purse." Iago intends, as he tells the audience at the end of the scene, to "make [his] fool [his] purse." Which means that he is convincing Roderigo to keep the money coming so that he, Iago, can make use of it.
Roderigo is an easy mark for Iago, a dupe who believes anything he says and follows his suggestions to the letter. Since Othello is presented as wise and well-spoken, Shakespeare has created a nice foil to him in Roderigo. In time, Othello will be dancing to Iago's tune just as easily as Roderigo.