Are names like Laquisha and Daquan African in origin or did they come from somewhere else?
Roughly speaking the rise of names you're talking about was around the time of the civil rights and black power movements in the US, and there are many influences. It's also important to remember that Africa is not monolithic. There's no one African culture just like there's no one European culture. There are many cultures on the continent that were influenced by contact, trade, colonization, and conquest from both inside and outside Africa.
Some come from African cultures. Monifa is Yoruba. Kwame is Akan.
Many Northern African cultures are largely Muslim and have been since the spread of Islam. Then there was the popularity of the Nation of Islam at the time of the civil rights movement. So you can get a lot of names of Arabic origin. Shaquille, for example, is a variant of Shakil. Jamal/Jamaal, Malik, Aisha/Ayesha, Aaliyah, Kareem, and Omar are other examples of names you might think of as African American that are of Arabic origin.
Others are French or take French influence. Monique, Andre, Antoine, Leroy, and Lamar are French.
Others come from further widespread cultures. Tyrone and Shawn are both of Irish origin. Tupac, as in Shakur, comes from Inca origins. Tamika is thought to come from Japanese Tamiko.
Still others are combinations of names. Jalen Rose's name is a blend of James and Leonard. Dashawn is adding the Da prefix to Shawn. Deonte is a blend of Deon (from Gideon, which is Hebrew) and Dante (Italian).
Then there's names that appear to have been invented fairly recently or are recent variant spellings. Not unlike the whiter phenomenon of names like Kynslee, Jaxxon, or Nevaeh.