Jane as a person did not really act independently. She was sixteen at the time of her death and had been used politically, rather than formulating political policy herself. As a person, she is a fascinating figure and the main sense of her that I have is of wasted potential: she could have been so much more had she lived and her party succeeded.
In spite of this, I do think that Jane had an influence on history due to what she embodied. Before the reign of Henry VIII, the succession had been, at least nominally, based on hereditary, with sons succeeding fathers, etc. Henry VIII, with his difficulties in obtaining a male heir, moved away from this. In the first two Acts of Succession of his reign, he disinherited each of his daughters in turn, based on the supposed invalidity of his marriages to their mothers. These marriages were arguably valid but Henry did, at least, base his attempts to change the succession on the principals of hereditary: illegitimate children had no right to succeed. The second Act of Succession made a real change however in that it allowed the king to name a successor if he had no legitimate heir: something which gave him the power to choose anyone that he liked. This was followed by the Third Act of Succession, which reinstated the two daughters, in spite of the fact that they were illegitimate. Henry VIII’s Will made the final change, bequeathing the crown to his legitimate son, Edward, and then to his two ‘illegitimate’ daughters, Mary and Elizabeth, before passing the throne to the heirs of his younger sister, Mary Tudor, ignoring the heirs of his elder sister, Margaret.
Henry VIII set a precedent in altering the succession and his son, Edward VI, followed this in his Device, passing over the claims of his two half-sisters, the heirs of Margaret Tudor and his cousin, Frances Brandon (the daughter of the elder Mary Tudor) in order to pass the crown to Lady Jane Grey. Had Jane secured the throne, this would arguably have cemented the precedent into law that the monarch had some ability to choose their own successor.
The failure of Jane’s attempt at the crown therefore did influence history and once again put the emphasis on the importance of strict hereditary title. Mary I hated her half-sister, Elizabeth I, but did not attempt to divert the succession from her. On her accession in 1558, Elizabeth I’s heirs were legally Frances Brandon and her daughter, Catherine Grey, due to the terms of Henry VIII’s Will. However, it was well known that Mary, Queen of Scots and, later, her son, James VI, had the strongest hereditary title. The succession in Elizabeth’s reign was always open to question and yet she always refused to name a successor, even on her deathbed in 1603. The fact that James VI of Scotland was able to succeed smoothly to the throne arguably has its roots in Jane Grey’s failed bid for the crown. In 1603 hereditary won over strict legal title. This was the same for Jane in 1553. It would be nearly the end of the seventeenth century, and after a civil war, before the succession again moved away from the strict hereditary line.