This is a really interesting question!

Probably the most crucial factor of her enduring popularity was her role as queen during World War II. It's not a controversial opinion, I don't think, to say that the home front experience during the 1940s had a massive effect on British culture that can still be felt to this day. It was common knowledge at the time that the queen was taking shooting lessons to defend Buckingham Palace if necessary and refusing to send her daughters to Canada, and she famously told the press that she was glad the palace was bombed so that she could look the East End in the face. The fact that the public saw her going out to talk to ordinary people in hospitals and bombed-out streets, even if part of a crafted publicity campaign, was a huge deal. Hitler called her "the most dangerous woman in Europe". In some ways, she embodied the concept of "Blitz spirit" that the British people often prided themselves on after the war, so it's not surprising that she would continue to enjoy residual acclaim from that. But that's not the whole of it.

Elizabeth Bowes-Lyon was actually fairly popular within the aristocratic set even before her marriage. Just a month after she debuted in society (at an enormous ball thrown by her family in 1918) she was being courted by one of the most eligible English bachelors, Charles, Lord Settrington, who was in line to be Duke of Lennox, Duke of Richmond, and Duke of Gordon, and others followed. The Hon. Stephen Tennant, one of the leading "bright young things" and the younger brother of one of her beaux, believed that she was quite shrewd and calculating in collecting suitors, but his comments can be taken in a more positive light to simply mean that she had a high social intelligence and was good at playing the game; the bright young things were very involved with artists, actors, and writers from the middle classes and deliberately flouted conventions, and would certainly not appreciate a woman who chose to pursue a "normal" upper-class woman's marital career. She was a good actress, and she was ambitious. In fact, she had wanted to marry the future Edward VIII, even while being courted by his younger brother. However, the Prince of Wales was not interested. Queen Mary had to help entice Elizabeth to stop stringing along her second son while holding out for the eldest.

As soon as she was engaged to Prince Albert, she gave a chatty, informal interview to the press, totally against royal protocol. Her in-laws-to-be weren't happy about it, but the reporters were charmed and introduced her to the public as their new sweetheart. She would go on to continue being the most popular member of the royal family in the years to come even though she would never again break protocol in this way, and would only use official appearances to be seen in the newspapers. She was Princess Di before Princess Di, but without the tragedy and scandal - the public could identify with her as an attractive former subject, and she knew how to appear to her best advantage. With a doting husband (whose retiring manner was a great foil for her) and two adorable daughters, in addition to her own engaging and flirtatious manner, she was a combination relatable and aspirational figure. The Duke and Duchess of Windsor would mock her for being dowdy and unelegant, so different from themselves, but it was her tendency to look like her subjects that helped them perceive her as a real human being.

You might enjoy the tell-all The Untold Story of Queen Elizabeth, the Queen Mother by Lady Colin Campbell. It's a much more candid view than you'll get in authorized biographies, but good reading.