Squeaky, the narrator of the story, is called Mercury by the big kids in her town because she is “the swiftest thing in the neighborhood.” Her real name is Hazel Elizabeth Deborah Parker, and people call her Squeaky because of her squeaky voice. However, she is also faster than everyone she knows, except for her father, and this explains her second nickname.
Mercury, in Greek mythology, is the messenger god, known for his extreme speed and swiftness. He has winged sandals that help him to travel at top speeds, delivering messages for Zeus, his father, and he flies all over Greece and Mount Olympus as required. For this reason, then, it makes sense that Hazel, or Squeaky, would be called Mercury by the older kids.
Mercury is also the name of the planet with the fastest revolution around the Sun in our solar system. It takes only eighty-eight days for the planet to complete one orbit, and it moves at almost twenty-nine miles per second, which is faster than every other planet of those that orbit our Sun. This is likely why it was named after the fastest Greek god.
So, we have lots of cultural associations with the name Mercury, and all of them have to do with speed. Thus, Hazel/Squeaky is nicknamed Mercury due to her ability to run so fast.