The character La Blanche in Kate Chopin's "Désirée's Baby" is important throughout the story. The name itself is significant, as has been said, because it literally means "the white." This entire story focuses on race, so the name is meant to indicate that this slave, whose cabin is near the main house, is one who could, given the right circumstances, pass for white. This ability to pass is extremely important for Armand, whom I would argue, is aware of his mixed heritage.
In the story, La Blanche is the mother of the "little quadroon" boy fanning the baby. It is in her looking back and forth between La Blanche's boy and her own son that Désirée first becomes aware of what so many others have already noticed: her son and La Blanche's son are the same color—they are both mixed.
Hidden or mysterious identity his suggested throughout the story. Désirée is first found in the "shadow or a column" and is of unknown parentage. It is not surprising that the mixed child might be attributed to her "obscure origins." But a closer reading will reveal that it is Armand whose surroundings are secretive. The very description of L'Abri, the plantation, indicates obscurity with its "roof [that] came down steep and black like a cowl, reaching out beyond the wide galleries that encircled the yellow stuccoed house." A cowl is the hood of a monk's robe and obscures his face. This "cowl" is black and hangs over the "yellow" house like a dark secret. Note that the nursemaid, Zandrine, is also called "yellow," a color that indicates mixed heritage. So if a dark secret covers a house whose color indicates mixed race, it suggests that what is being covered is the family's mixed race.
Some have argued that Armand only learned that his quadroon child was a result of his own mixed race after the presumed death of his wife and baby, but I would argue that he knew all along and was hoping for a child that was white enough to pass—a child "as white as La Blanche." This knowledge would explain his rush to marry Désirée, whose shadowy ancestry could be used as an excuse should he father a child who was "not white." He could blame his wife, who, too, was "as white as La Blanche."
The fact that La Blanche lived so conveniently close to the main house, that Armand frequented her cabin, and that she had children might suggest that Armand was her son's father. Her son, a quadroon, is darker than La Blanche. Armand, as Désirée points out when Armand accuses her of being "not white," is also darker than Désirée, who Armand retorts, is "as white as La Blanche."
La Blanche, the whitest slave, represents a person of mixed race who is white enough to pass for white. Had her heritage been unknown, like Armand's and Desiree's, she could have passed for white.
But since the child did not pass, Armand was forced to blame his wife. He was forced to burn the last remaining trace of his own heritage, the letter from his mother. I find it difficult to argue that he was unaware of this letter, which he kept in the same drawer as his letters from Désirée. I also find it difficult to argue that he, who was eight years of age when his mother passed away, was unaware of his heritage, unless, she, too, was "as white as La Blanche," belonged "the race that is cursed with the brand of slavery."
Your question refers to a character who is mentioned three times in Kate Chopin's story "Desiree's Baby". La Blanche is one of the slaves at the L'Abri estate where Desiree lives with her husband and infant. The first time La Blanche's name is mentioned is when Desiree described how loud her baby can cry; it can be heard "as far away as La Blanche's cabin". The fact that La Blanche has a cabin is a clue to us that she is probably one of the slaves.
The second time her name is mentioned is when Desiree's baby is three months old and "one of La Blanche's little quadroons" is doing the job of fanning the baby. La Blanche, then, is implied to be the mother of a child that is one-quarter black. If her child is one-quarter black, it must mean that his mother is half-black and his father is all black. La Blanche, as a half-black and ostensibly half-white person, may have been named "La Blanche" because her name means "the white" and because her skin was likely paler than the other slaves' skin color due to her heritage.
The third time La Blanche's name is mentioned is when Armand speaks cruelly to Desiree as she desperately tries to get him to agree that Desiree is white, even whiter than Armand himself. He replies that her hand is "as white as La Blanche's". Armand means that Desiree is only as "white" as a half-white person, and therefore is basically black in his estimation.
The irony of this, of course, is that we later learn that Armand himself is the one who had a black parent. We never learn anything about Desiree's true parentage. For Armand to use a comparison to La Blanche as an insult would be the height of hypocrisy if he had known that he too had one black parent. Unfortunately, Armand's belief that Desiree's bloodline is responsible for their baby's darkening complexion leads to Desiree's desperate decision to leave. We can only suppose that she and her baby have perished in the cold October forest, having left in clothing inadequate for the season. Armand does not discover the truth that his mother is black until after Desiree's disappearance.
The name "La Blanche" serves a primarily symbolic purpose. It means "white", and is thus directly connected to the issue of La Blanche's parentage. Furthermore, white is often used as a symbol for purity or innocence. When Armand compares Desiree to La Blanche in terms of color, it may be an attempt by Kate Chopin to say something about Desiree's (not to mention La Blanche's) position of being an innocent victim of racial prejudice. Desiree's clothing is white, she lies upon white muslin, and she is as white as a stone statue when waiting for Armand to read her mother's letter. The white imagery that runs throughout the story indicates that there is a question of virtue at stake in it.
"La" is a feminine article, and "to blanch" means to make white or pale by extracting color; to bleach. Example: Michael Jackson blanched his skin. So, "La Blanche" literally means "the white female." At first, we think it refers to the baby, but as we uncover Desiree's mother's secret, the white one refers ironically to Desiree herself. In other words, Desiree is not entirely white.
Desiree's baby is working in the opposite direction of Michael Jackson; it is attaining color instead of extracting it. The baby is born white, and then it begins to attain color as the pigmentation sets in. The baby begins to look brown, or mulatto over time, and rumors that Desiree fathered the child with a black man are spread around the plantation. The name is ironic because Desiree expects the baby to be white because she is white and the father is white, but her mother knows that Desiree is not entirely white.
White also has connotations according to the color code: "white is right"; "white is might"; white is blameless, pure. By contrast, black is symbolically corrupt, evil, etc... So by the end, the reader and Desiree realize the irony of the situation, that although Desiree is blameless for having relations with a slave (her mother is the guilty one), Desiree and her child will be blamed for it. The mother and child have violated the color code; the mother will never be blamed.