I think we are dealing with two distinct families here: the Scottish Grays and the English Greys.
Lady Jane Grey's claim to the English throne was through her maternal grandmother, Mary Tudor, the younger sister of Henry VIII. Henry's third Act of Succession left the throne to his son Edward, then his daughters Mary and Elizabeth. If none of his children had issue, the crown went to the descendants of his sister Mary. Henry purposely excluded the descendants of his older sister Margaret (married to James IV of Scotland) because he didn't want the crown to pass to the Scots!. When Henry died the crown passed to the 10 year old Edward who only lived for five more years. His will excluded his sisters and named Lady Jane Grey as his successor. Whether a fifteen year old could write a legal will or whether the will could nullify an Act of Parliment is a moot point. The nation preferred a daughter of Henry to his grand niece and Lady Jane paid with her head.