I understand that this may have been an attempt to relinquish the responsibility of choosing between his siblings but would not electing a relatively unknown claimant of the throne who lacked any serious support only jeopardise the stability of England further? Not to mention that his father, Henry VIII would likely have much preferred the crown to pass to Elizabeth than a distant relative.
As Edward did it while dying, we don't know his exact motives. He was likely influenced by his advisor Northumberland. But, in short, Mary (the older) and Elizabeth (the younger) were originally considered bastards, until restored by the Succession Act of 1544 (and the passing of that act indicates that yes, Henry probably would have preferred it to pass to his daughters). But Mary was still older than Elizabeth. There was no legal reason to argue that Elizabeth should go ahead of Mary. By skipping both, the argument Edward's advisors could use is that Edward had simply reversed the 1544 Act. Obviously, the goal was to get a Protestant onto the throne, not a Catholic. But Elizabeth's claim was, simply, the same as Mary (daughter by a repudiated wife), but worse because she was younger.
This inheritance occurred during the ongoing religious disputes in England, between the nascent Church of England (created by Edward VI's father) and the Catholic church. Both the nobility and the population were split on the matter, with powerful and potentially violent factions on both sides.
Edward VI was the son of Henry VIII's third wife, Jane Seymour. Mary and Elizabeth, his half sisters, were from two different wives, Catherine of Aragon and Anne Boleyn respectively. Both had been repudiated and their offspring alternately removed and restored to the succession over time. They also were from opposite sides of the religious dispute. Mary was a staunch Catholic (Catholicism being an important part of her mother's and her own claim to continuing legitimacy) and Elizabeth was a protestant (an equally important part of her mother's and her own legitimacy).
Edward VI was himself devoutly protestant, and wanted the throne to go to a protestant after him. His only really effective means of skipping Mary was to declare that her mother's marriage to their mutual father Henry was invalid. Doing so would put him in the corner of having to declare Elizabeth's mother's marriage to Henry equally invalid, since the same arguments applied to one as the other. This meant that if Mary couldn't inherit, neither could Elizabeth.
There were no other children of Henry VIII, so the next place to check following the laws of succession in England were to the lines of Henry's siblings. The older sister's line were Catholic, but the younger sister's line was Protestant. The eldest member of the new generation in this Protestant line was Jane. Jane was married to Lord Dudley, who happened to be the son of Edward's powerful advisor, the Duke of Northumberland. It has been suggested that the Duke may have pressured the dying young king in to "realizing" that Jane was the legitimate non-Catholic successor.
Edward VI was a Protestant, raised under the Church his father Henry VIII had established after he had broken with Rome. According to the Act of Succession 1543, Mary was to follow Edward if he didn't have any heirs, and then if Mary didn't have any heirs, Elizabeth was the next in line. Mary was a devout Catholic and opposed to the reforms her brother had passed. Edward wanted a Protestant successor and Mary didn't fit the bill. In addition both of his sisters were declared illegitimate by his father which lessened their suitabilities as heirs, so Edward chose Lady Jane Grey to be his successor.