The influence of Islam on African-Americans during the Civil Rights era is a bit complicated. Islam has been and is, undisputably, a major religion in various African countries right up to the present day. However many of the more media-focused "Islamic" groups arrived at the faith in a sort of complicated fashion.
In 1913, Noble Drew Ali founded the Moorish Science Temple of America. Ali claimed to have traveled widely and been initiated and/or recognized by an Egyptian priest and given a "lost" section of the Qur'an, which Ali then worked into Holy Koran of the Moorish Science Temple of America (this book actually contains borrowings from 1907 work The Aquarian Gospel of Jesus the Christ and the Rosicrucian work Unto Thee I Grant, probably the 1925 edition). One of the characteristics of Ali's teachings is that Black Americans were not actually "African" in the sense of sub-Saharan Africa, but were actually Moors and that their native religion was Islam. This was a very important point given the legal and racial discrimination that African-Americans suffered during the 20th century: the Moorish Science Temple provided an outlet for African-American identity outside of the strict confines of black and white.
The Moorish Science Temple split into schisms after the death of Ali in 1929. One of the members who broke away was Wallace Fard Muhammad, who founded the Nation of Islam in Detroit in 1930. Fard disappeared in 1934, and the Nation of Islam was taken over by Elijah Muhammad, who held control of the organization until 1975.
Like the Moorish Science Temple, the Nation of Islam promoted racial identity strongly - although it did not go so far as to claim African-Americans weren't "black," it did tailor its message and theology specifically toward Black Americans, in books like Elijah Muhammad's Message to the Blackman in America (1965). So the Nation of Islam's approach to Islam is somewhat like the Church of Latter Day Saints approach to Christianity: definitely related (one might say derivative) but skewing its own way. The Nation of Islam rose to national prominence during the Civil Rights era, especially with charismatic members such as Malcolm X, Muhammad Ali (who changed his name in 1964 at Elijah Muhammad's suggestion), and Louis Farrakhan. Malcolm X would later leave the Nation of Islam after internal and theological disputes, and convert to Sunni Islam.
The Nation of Islam had a very widespread cultural appeal because of its strong black nationalist message during a period of rising attention to segregation and African-American Civil Rights, and it also influenced the creation of related groups and faiths. One such outgrowth is the Five-Percent Nation, founded in Harlem in 1965 by Allah the Father/Clarence 13X. Like the Nation of Islam, this is an Islamic-influenced theology with added mystical theology such as the Supreme Mathematics and the Supreme Alphabet. Five-Percenter theology has an influence on popular culture, especially hip-hop, such as the works of the Wu-Tang Clan, Jay-Z, Rakim, Naz, Big Daddy Kane, etc.
The story of Islam and its influence on African-Americans is much more complicated than that; there were many different Muslims that immigrated to the United States, many converts to various sects of Islam, etc. But the real popularization of Islam in the United States owes much to these Islam-influenced religious organizations which focused on Black nationalism at a time when Black nationalism and civil rights was becoming a major social issue in American life. High-profile events like Cassius Clay becoming Muhammad Ali and the assassination of Malcolm X really raised the profile of Islam as a potential way for African-Americans to assume a distinct identity away from white power structures.
For more on this topic in general, I would reccomend A History of Islam in America: From the New World to the New World Order by Kambiz GhaneaBassiri.