In Pygmalion, what is Liza's one quality that most fascinates Higgins and Pickering?

Higgins and Pickering are fascinated by the way Liza speaks.

Higgins is a scientist of phonetics. When he meets Liza in the rain, he jots down her words phonetically and then guesses her exact birthplace from her accent.

Pickering is a linguist of Indian dialects. Pickering is on his way to introduce himself to Professor Higgins when they meet by accident on the corner in the rain. Pickering observes Higgins recording Liza's words and guessing her origin. He finds Higgins's ability to pinpoint a person's birth place from their accent admirable.

Later, Higgins and Pickering are socializing at Higgins's house when Liza comes and offers to pay a shilling to Higgins to teach her to talk like a lady. Pickering offers to pay for the lessons under the condition that Higgins can pass Liza off as a duchess at an upcoming garden party. Higgins therefore is initially interested in Liza for her cockney accent, while Pickering is interested in whether Liza can change her accent and learn to speak like a duchess.