Why do you think "La Blanche" had that name?
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.