In the lengthy article about Lady Arbella in Isaac D'Israeli's Curiosities of Literature, he explained that, like the other crowned heads around her, the Pope also had matrimonial plans for Arbella: He wanted her to marry a member of the House of Savoy (descended from one of Edward IV of England's many bastards) and then take the English throne, leaving James king of only Scotland (which would make the king of France happy). But the duke of Parma was already married, so the Pope ("in his infallibility," says D'Israeli) settled on his own brother, who was a Catholic Cardinal. To eliminate that impediment, the Pope defrocked his brother, freeing him to marry "Arbelle" (as the Italians spelled her name) and thus become king of England. Of course no more came of this notion, either, and it may be superfluous to mention that nobody, then or now, has ever been sure Arbella was a Catholic -- she was flexible enough that everyone who talked with her thought her to be of the same religion as they were.
Neither would King James allow her to marry, once he was king of England and had children of his own, because her children would have had a claim to the throne. In 1603, right after James came to the English throne, there was a plot (in which Sir Walter Raleigh was involved) to overthrow him and put Arbella on the throne, but when she was invited to participate by writing to the king of Spain agreeing to it, she reported it to James instead, and that was the end of that. In 1604 the king of Poland sent an ambassador to England to ask for Arbella to be his queen, but D'Israeli could not discover whether it was James or Arbella herself who rejected that offer.
There are some indications that she tried to elope in about 1604 and fell out of favor with King James because of it; she was certainly out of sight until 1608, when she was restored to his good graces. In 1610 she did elope and marry William Seymour, 2nd Duke of Somerset, grandson of Lady Catherine Grey, a younger sister of Lady Jane Grey. For that, King James imprisoned them: Arbella in Sir Thomas Perry's house in Lambeth and Seymour in the Tower of London; they had some liberty within those buildings, and some of her letters to him and to the king during this period survive. When the king found out about her letters to Seymour, however, he transferred Arbella to the custody of the bishop of Durham, but Arbella claimed to be sick, so her departure for Durham was delayed.
The couple used that time to plan their escape. She dressed as a man and got away to Lee (in Kent), but Seymour did not meet her there before their getaway ship had to sail for France. Seymour did escape from the Tower, but by the time he got to Lee, she was gone, so he caught the next ship he could and sailed to Flanders. The ship Arbella was on was overtaken by King James's men just before it reached Calais (France), and she was taken back and imprisoned in the Tower of London. She never saw her husband again, and she died in the Tower in 1615.
In 1993 a collection of more than a hundred of Arbella's letters was published, edited by Sara Jayne Steen, providing details of her activities and ideas.