"I kind of felt like it was meant to happen, to have her on this day," the mother said.
Eclipse was born at Greenville Memorial Hospital, measured 19 inches long and weighed 6 pounds and 3 ounces, hospital spokesman Dudley Brown said.
The family originally planned to name the baby girl Violet, but decided to give her the name Eclipse in celebration of the rare event.
South Carolina was one of 14 U.S. states in the path of totality for Monday's solar eclipse, and Eubanks said while her family was "shocked" believed it was destiny.
"I think it was just meant to be, her name," she said. "We're probably going to call her Clipsey."
About 11 other babies were born on Aug. 21 and were given special onesies to celebrate the occasion.