Adam Clarke on Luke 1:36

Thy cousin Elisabeth - Thy kinswoman, συγγενης. As Elisabeth was of the tribe of Levi, Luk 1:5, and Mary of the tribe of Judah, they could not be relatives but by the mother's side.

She hath also conceived - And this is wrought by the same power and energy through which thou shalt conceive. Thus God has given thee a proof and pledge, in what he has done for Elisabeth, of what he will do for thyself; therefore, have faith in God.

Who was called barren - It is probable that Elisabeth got this appellative by way of reproach; or to distinguish her from some other Elisabeth also well known, who had been blessed with children. Perhaps this is the reproach which Elisabeth speaks of, Luk 1:25, her common name among men, among the people who knew her, being Elisabeth the barren.