"But she named the infant "Pearl," as being of great price-purchased with all she had,-her mother's only treasure!" Chapter 6
Hester paid the price for her "sin." Naming her child Pearl seems ironic, This is Hester's way of letting everyone know although she admits to her sin, she is also not ashamed but proud of it.
Do you really think that Hester is entirely proud to have Pearl as her daughter? Certainly, upon looking at later points in Chapter 6 and how oddly the child acts, even in her mother's presence alone, it doesn't really seem to come across as Hester being completely boastful that this is her daughter. Sure, Hester may dress Pearl in the same fashion and colors as the insignia that she wears on her clothes to label her an adulterer, but is that really a way to show pride toward this unique child? Hester even doesn't seem quite so sure about whether or not Pearl is her daughter in the first place with all of her impish and off-the-wall behavior. This is seen toward the end of Chapter 6 when Hester asks Pearl, " "Child, what art thou?" " or, " "Art thou my child, in very truth?" " Although a few lines later, Hawthorne does note that Hester is "half playfully" telling Pearl that she isn't her daughter, there is still the matter of that other "half" in the "half playfully." Does the other half actually think that Pearl is simply a demonic manifestation of her sin that has come to life to torture her?
Hester may not be enitrely proud of her daughter, but rather than show shame and guilt, Hester makes Peal's existence obvious. When you compare Pearl to the other Puritan children, of course she appears to be "off the wall." Maybe she was ADHD. As a mother, there were times I wondered if "thou art my child?" (Of courswe they are, but things they do can make you wonder.)
Maybe a small part of Hester really does wonder if Pearl is a "demonic manifestation of her sin that has come to life to torture her", but like Valerie said, Hester would never want to show society shame or guilt towards her. The last thing Hester wants is to feel like society has won over her.