[Ibrîniðilpathânezel]

Since in a sense the wizards were "stewards" of Middle-earth, as Gandalf tells Denathor, my personal interpretation has been that they are largely symbolic to their office, as the steward of Gondor has a white rod signifying his position. Whatever the case, we do know that Gandalf wasn't the only one with a staff, as Saruman had one that was broken when he was cast from the order. And if Saruman is to be believed, all five Istari had staves, since when he accuses Gandalf of wanting greater power, he refers to the staffs of the Five Wizards. To me, it makes sense that they are emblems of their office, as Gandalf breaks his to shatter the Bridge as his own office comes to its moment of personal failure, and Saruman's is broken when he loses his position as the head of the order and is cast from it. But there is no concrete proof for my theory, only conjecture partially based on my perception of a vague similarity in symbolism between the staff of a wizard and the signet of the Gondorian steward.