The most obvious significance of Beneatha's name in the play (and perhaps the only significance of the name) is the idea that it is not a "white" name.
Beneatha's character is largely defined by her troubled sense of identity and her striving after a dignity sourced both within her genetic history, as it were, and outside of her home/living conditions. Beneatha is looking for ways to be proud (or ways to legitimize her natural sense of pride) and she seeks this pride in a Pan-African philosophy (embodied by Asagai).
"Proud of her African heritage, she is inspired by the attentions of a Nigerian suitor to wear an Afro instead of a processed, straightened hairstyle" (eNotes).
In attaching identity to Pan-Africanism, Beneatha enacts the ethnicity of her name, in a manner of speaking. She intentionally and consciously distances herself from the culture of white oppression (and "whiteness" in general) while also notably stepping away from a sense of being oppressed.
Also, Beneatha wants to gain stature in her professional life. This ambition relates to race and to overcoming racial bias/limits/discrimination, but it also relates to gender bias. Beneatha refuses to be oppressed by any dominant group - or by any individual, for that matter. She will not be dominated by her brother or her mother, by gender expectations or by racial discrimination.
At one point she expresses her independence in regard to generic expectations while talking to Ruth and Mama.
"Listen, I'm going to be a doctor. I'm not worried about who I'm going to marry yet - if I ever get married."
While we may be tempted to suggest then that Beneatha's name becomes ironic because of her insuppressible mentality (refusing to be "beneath" anyone), we should tread cautiously around such interpretations. What we can say with confidence is that the name "Beneatha" seems to be a "black" name.
If this is the case, we might see Walter's name as having a double significance. He is named after his father and so derives his identity in part from the family. His name is not as ethnically specific as Beneatha's, however, and so we may read his particular dream of business ownership against Beneatha's dream and wonder if Walter is meant to be seen as participating in the American system in ways that Beneatha would reject.
Interpretations like this are highly conjectural. We can only say these meanings may exist as intentions of the author.
As a side note, Lorraine Hansberry has said that Beneatha's character is based on her, the author of the play. Hansberry's ambition and activism are integral aspects of the Beneatha character. This anecdotal information may speak to a reason to limit the scope of what we project as to the meaning of Beneatha's name.
Beneatha often acts like everyone is below her or beneath her. Her superior attitude comes from having more education than the rest of the family and her desire to be a doctor. She challenges Mama with the statement,"There is no God". Mama quickly rises to the challenge by slapping Beneatha and having her repeat, "In my Mama's house, there is God." She also badgers Walter by saying, " I dissected something that looked just like you the other day." Beneatha is constantly experimenting with different "forms of expression" to discover herself and often overlooks the facts that Mama, Walter and Ruth have worked to help put Beneatha through school. This attitude makes it hard for others to live with her, until she meets Asagai, who seems to understand her and he helps her be more tolerant and see things from other people's point of view.