How is the name of Hester's child symbolic in "The Scarlet Letter"?
There are religious meanings in this book that focus on sin and suffering and the choices that people make and how it affects their eternal salvation.
In the Puritan theology, or beliefs, people are either saved, intended for salvation or they are not. Salvation would be very important to all Puritans including Hester and Arthur.
The name Pearl is symbolic, as in pearl of great price. This reference can be found in the Bible, Matthew 13:45-46.
"Even the use of "seeking" (Matthew 13:45) helps to identify the merchant as Christ because it means "to depart from one place and arrive at another." Jesus did this Himself to pay the price of the pearl. He departed from heaven and arrived on earth to complete His mission."
The child, Pearl, has brought Hester to a new place, her salvation is in question.
Pearl, her daughter, has cost Hester dearly, including, possibly her chance at salvation.
Hester probably considers Pearl beautiful and rare, just like real pearls. Puritans also considered pearls as a luxury and vanity. Hester and Dimsdale sinned when they had an affair, and Pearl is the result of that luxury and vanity.
Pearl are created by an irritant entering an oyster, usually a piece of sand or grit. Hester was the irritant thrust into the oyster that was the Puritan society. She "spoke out" via her affair with Dimsdale. The result is Pearl.