In the story, Monsieur Armand Aubigny fell in love with Desiree when he spotted her standing against the stone gateway of the Valmonde estate.
Accordingly, Desiree was adopted by the Valmondes as their daughter after she wandered onto their property eighteen years before. At the time, no one had known where the toddler had come from. There was speculation as to her origins, but in the end, Madame Valmonde had concluded that Desiree had come to her as a gift from God, 'seeing that she was without child of the flesh.'
When Monsieur Armand Aubigny asked for Desiree's hand in marriage, Monsieur Valmonde had been wary. He raised the issue of the 'girl's obscure origin' to Monsieur Aubigny. But, the author tells us that Monsieur Aubigny did not care:
He was reminded that she was nameless. What did it matter about a name when he could give her one of the oldest and proudest in Louisiana? He ordered the corbeille from Paris, and contained himself with what patience he could until it arrived; then they were married.
The phrase 'he was reminded that she was nameless' alludes to Desiree's 'obscure origin.' No one knew anything about her status or her family history; therefore, she had no family name, in the sense that no one could ascertain whether she came from an impoverished, middle-class, or wealthy background. Because Desiree had no established background, Monsieur Aubigny considered her 'nameless.'
However, he was not perturbed by this lack on Desiree's part; he reasoned that he would be able to give her 'one of the oldest and proudest' names in Louisiana when he married her. In giving Desiree his family name, Monsieur Aubigny felt convinced that Desiree's status in society would be raised.