Why is Pearl called a "red rose"?
Because roses are beautiful but have thorns. So too does Pearl. She is physically beautiful but has a mean streak in her. She is called a "red rose" by Dimmesdale, who thinks it complimentary, for the child is so beautiful. But Dimmesdale soon discovers the thorns: Pearl “threw one of the prickly burrs at the Reverend Mr. Dimmesdale. The sensitive clergyman shrunk, with nervous dread, from the light missile”
Pearl is often cruel to her mother as well. She enjoys edging the scarlet "A" with the irksome burrs: "arrange[ing] them [prickly burrs] along the lines of the scarlet letter that decorated the maternal bosom."
She is a "red" rose in particular because she is tainted by proxy through her mother's "sin" and also probably red because she is a flesh and blood human being who does not hesitate to show her anger.
Hawthorne is also using this epithet to connect readers back to the first chapter, The Prison Door. Pearl is like the red rose bush that grows next to the prison. The prison represents the sin of society, and it represents failure. It is weathered and clearly aged. It is like the adults in the Puritan community, set in their ways, their prejudices, and their sins. Pearl, however, is an innocent, and in the end breaks free of her community, thrives, leads a happy and successful life. She is the rose that brings hope to the prisoners of the past. Hawthorne chose very carefully to write of an earlier time, to comment on his own ancestors and the sins that they committed, but uses Pearl to demonstrate that those sins have, or can be, overcome.