A number of sources discuss the evidence in favor of Elizabeth speaking the Magnificat, though most of them ultimately reject it as insufficient. Here are the basics; refer to the references below for fuller treatment.
First, a few manuscripts and church fathers have Elizabeth, rather than Mary, in Luke 1:46. On this basis A. von Harnack and others have suggested that the original text had "and she said," not identifying the speaker, only later did guesses get added to the text. If this is accepted, then the evidence for Elizabeth mounts:
- Elizabeth is the last speaker before 1:46, and "Mary" is explicity identified in v. 56, suggesting a new speaker
- The Magnificat has obvious similarities to Hannah's song of 1 Samuel 2, and Elizabeth and Hannah shared the condition of barrenness
- The parallels between the Magnificat and the Benedictus given by Elizabeth's husband, Zechariah
That said, scholars still find this evidence insufficient in light of the counterarguments to these arguments1 and especially when considering the vast textual evidence supporting the "Mary" reading in Luke 1:46. Stephen Farris writes:
That Luke attributed the Magnificat to Elizabeth has been, as we have seen, argued with considerable learning and ingenuity. In my judgment it must, however, be rejected. The positive arguments in its favour can be countered and there remain certain important considerations which suggest that the hymn be attributed to Mary.
Taken together with the near unanimity of the textual witness, these arguments can give one considerable confidence that the hymn belongs to the one to whom it has been attributed 'from of old.'
1 Emmet and Farris in particular devote significant effort to dismantling the arguments summarized here.