On Facebook, Christopher Rollston posted this article about the Pope and Israeli Prime Minister Benjamin Netanyahu getting into a disagreement about what language Jesus spoke. When Netanyahu claimed that Jesus had spoken Hebrew, the Pope interrupted and said he spoke Aramaic. According to the article, Netanyahu conceded that Jesus spoke Aramaic but that he knew Hebrew as well.
In terms of the state of the discussion, Pope Francis is right. Most Jesus scholars now agree that Aramaic was Jesus' everyday language, though he may also have had some facility in Hebrew and possibly even Greek. Stan Porter has argued for the Greek side of things. All of these languages, as well as Latin and Nabatean, are attested from Jesus' time and locale. The endlessly complex issue, however, is the degree to which any given instance of one of these languages is indicative of everyday realities, and further whose "everyday" we're talking about, since things would have been very different for, say, someone in rural Galilee, someone in the Jerusalem temple or at Qumran, and someone in Pilate's house when he's come to town from Caesarea Maritima.
What's perhaps most interesting about this question, however, is that it has a seriously long and detailed history of research. Although not all of it is motivated for this reason, most of it is motivated by a search for the original words of Jesus in the Gospel texts. For the most recent statement on languages in Jesus' milieu, see the excellent article by John C. Poirier, friend of the Jesus Blog: "The Linguistic Situation in Jewish Palestine in Late Antiquity," Journal of Greco-Roman Christianity and Jduaism 4 (2007): 55-134. To my knowledge, this is the most thorough treatment since Fitzmyer's famous article and confirms Fitzmyer on a number of issues.