There are two major sources generally for name frequency in the United States: the Social Security office and the US Census. The Social Security data does not include racial identification, and while the US Census has released name data connected to race, it's only for surnames and only for 2000 and 2010.
There's a recent paper by Konstantinos Tzioumis which uses mortgage application data to extract first name frequency connected to race, but applies only to 2007. If you're curious, Tyrone was used with white people in approximately 32% of the US population and with black people in 59.5%.
There are other ways to get at data -- some recent papers pulled together multiple sources, including New Orleans Sales Data to track blacks names by popularity; for pre-20th century, the first names commonly associated were:
Abe, Abraham, Alonzo, Ambrose, Booker, Elijah, Freeman, Isaac, Isaiah, Israel, King, Master, Moses, Pearlie, Percy, Perlie, Purlie, Presley, Presly, Prince, Titus
(These are not the associations now, which means we shouldn't be going back into very deep history to look for our answer.)
Because of all this, it's unfortunately not possible with currently accessible data sets to completely answer your question, but we can at least make a guess based on the data we have.
You can get more exact numbers from the Social Security site by using the Popularity of a Name search feature.
The name "Tyrone" shows up no more than a blip until 1937. Then it has another jump between 1968 and 1969, going from a rank of 180th to a rank of 137th (you can see the same jump on the graphical data chart).
Now, we know that naming spikes quite often comes because of famous people or even fictional characters. For example, between 1990 and 1991 there is an enormous leap in the name "Ariel", which might seem puzzling were it not for the November 1989 release of the movie The Little Mermaid.
So going back to Tyrone, we have two events, one in 1937 (or a little before) and one in 1969 (or a little before). What happened? In both cases there really is only one person who fits the bill.
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Tyrone Power had his first leading role in 1936 (Lloyd's of London) and was, for a time, Fox's biggest star. He played Jesse himself in the 1939 Western Jesse James; he also played the title role in The Mask of Zorro (1940). Co-star Basil Rathbone commented:
Power was the most agile man with a sword I've ever faced before a camera. Tyrone could have fenced Errol Flynn into a cocked hat.
Arguably his most terrific performance came near the end of his life, with Witness for the Prosecution (1957) where he co-starred with Marlene Dietrich and Charles Laughton, but unfortunately he died of a heart attack the next year when filming Solomon and Sheba.
Tyrone Power was the great-grandson of Tyron Power, Jr, a famous Irish actor of the 1830s. So the name passed from Ireland due to a sheer fondness for it, and only because a popular baby name in the US the year after Tyrone Power became a Hollywood star.
Unfortunately, with the lack of race-linked data, we can't say with certainty if somehow the actor's fame was linked to black audiences (as opposed to all races generally). There isn't any evidence for this; he starred in fully mainstream Fox movies (and was even asked to be lent over for Gone With the Wind, but Fox refused). He did perform in the stage play John Brown's Body (1952-1953) but it involved a rotation of parts reading the poem by Benet and would not have had any particular reach to black audiences.
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The singer Tyrone Davis, on the other hand, is a much better candidate. His single Can I Change My Mind (1968) made #1 on the US R&B chart, and he followed with Is It Something You've Got (1969, not as big a hit, but still #5 on the R&B chart) and his most famous song, Turn Back the Hands of Time in 1970.
In 1969 he listened to Black Panther leader Fred Hampton (the one from the recent Judas and the Black Messiah movie) when he showed up to speak where Mr. Davis was performing. He was impressed and joined a benefit fundraiser after Hampton's murder; in his 1973 song Writin’ on the Wall he called for pan-African unity and sang about Malcom X.
Tyrone Davis's fame corresponds exactly to the "second spike" of the name Tyrone and is most likely the reason for the black association, as he was associated not just with black identity but the entire "soul music" wave of the 1970s.
There’s still a link between the two men; Tyrone Davis was born in 1938, meaning in all likelihood his Tyrone naming would not have happened without the other.
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Cohen, A. (2019). Move On Up: Chicago Soul Music and Black Cultural Power. University of Chicago Press.
Cook, L. D., Logan, T. D., & Parman, J. M. (2014). Distinctively black names in the American past. Explorations in Economic History, 53, 64-82.
Logan, T., Cook, L. D., & Parman, J. (2020). The Antebellum Roots of Distinctively Black Names (No. w28101). National Bureau of Economic Research.
Tzioumis, K. (2018). Demographic aspects of first names. Sci Data 5, 180025. https://doi.org/10.1038/sdata.2018.25