Albert Barnes Commentary Luke 1:36

"And behold, Elisabeth thy kinswoman, she also hath conceived a son in her old age; and this is the sixth month with her that was called barren." — Luke 1:36 (ASV)

Your cousin Elisabeth, etc. The case of Elisabeth is mentioned to inspire Mary with confidence, and to assure her that what was now promised would be fulfilled. It was almost as improbable that Elisabeth should have a child at her time of life, as it was that Mary should under the circumstances promised.