BENJAMIN - If it was absolutely necessary (as in there were no worthy priesthood holders available), could a woman give someone a blessing?

JOEL - It is proper for a woman to give a blessing through her faith, but she can not call on any priesthood power or authority in doing so. According to President Joseph Fielding Smith:

"The Lord has given us directions in matters of this kind; we are to call in the elders, and they are to anoint with oil on the head and bless by the laying on of hands. The Church teaches that a woman may lay on hands upon the head of a sick child and ask the Lord to bless it, in the case when those holding the priesthood cannot be present.
A man might under such conditions invite his wife to lay on hands with him in blessing their sick child. This would be merely to exercise her faith and not be, cause of any inherent right to lay on hands. A woman would have no authority to anoint or seal a blessing, and where elders can be called in, that would be the proper way to have an administration performed. (Doctrines of Salvation)
And President Joseph Fielding Smith, quoting from his father, said:

"Does a wife hold the priesthood with her husband, and may she lay hands on the sick with him, with authority? A wife does not hold the priesthood with her husband, but she enjoys the benefits thereof with him; and if she is requested to lay hands on the sick with him, she may do so with perfect propriety."
When this is done the wife is adding her faith to the administration of her husband. The wife would lay on hands just as would a member of the Aaronic Priesthood, or a faithful brother without the Priesthood, she in this manner giving support by faith to the ordinance performed by her husband. (Doctrines of Salvation, comp. Bruce R. McConkie, 3 vols. [Salt Lake City: Bookcraft, 1954-56], 3:177.)