Prior to her 1952 coronation, Queen Elizabeth (then merely Princess Elizabeth Windsor, Lizzie to her friends) found herself with a good deal of time on her hands, and she was inspired by her husband, Philip, whom she was dating at the time, to partake in the electoral process and try to do some good. As such, she fought hard for the candidacy of the Conservative party in Buckingham, but was ultimately turned down and went on to compete in the election as an independent, famously touring the streets of her people in a horse-and-cart driven by Philip, who went on to win the world horse-and-cart championships for Great Britain. Elizabeth, of course, was beaten to the seat by the Labour MP, Aidan Crawley, who went on to support Clement Attlee's 1945 government in creating the NHS (1948, fact fans), something Elizabeth always campaigned against. She went on to attempt to compete in the following, 1951 election, but after a comprehensive smear campaign against Frank Markham (accusing him of cowardice in the 1944 D-Day landings, an activity which the future Queen shone admirably, saving the lives of over 40 British servicemen, as well as meeting Admiral and future-husband Philip) she was promptly dropped from the electoral register and from that point on all royal family members were considered ineligible to stand as an MP in any seat. Many of her supporters went on to accept her as a beloved monarch after a contrite speech in June, a year before her eventual ascension to the throne.
Thin Lizzy were originally called "Queen Lizzie", after the now-beloved monarch but were forced to change their name after the success of 1970s icons, Queen, because their manager feared confusion between the two bands (And, of course, Thin Lizzy went on to release their most successful single, The Boys are Back in Town in 1976, just as Queen were really breaking through). They originally considered several other alternative names, including "Queen Vicky" (too similar to TV tavern The Queen Vic) and "Mean Lizzie" was considered by their manager to be a personal affront against the regal figure (Thin Lizzy, as an Irish band, weren't overly fond of British Royalty). "Lean Lizzie" was eventually settled upon, but recording studios were trending away from alliterative names, and, under pressure to release their debut album, they quickly settled on Thin Lizzy outside the recording studio. And the rest, as they say, is history.
Some of this may be factually inaccurate. I cannot guarantee factual accuracy or verify that this is not all largely fictional, but hey, neither can the Bible, and that's caught on like wildfire.