So, the Nigerian Prince scam is a type of advance-fee scam.
This type of scam goes back a very long time. In the late 1800s and early 1900s a common and famous example of this type of scam was the "Spanish Prisoner scam", where the scammer wrote letters to the mark, making the claim that a distant and unknown relative of the mark was quite wealthy, and was unjustly imprisoned in Spain. If the mark would just remit money for some fees, the distant relative would be sure to provide ample reward. Here is a story from the Minneapolis tribune from 1910 warning about the scam.
Now, why is the Nigerian Prince scam known as the Nigerian Prince scam?
The answer is pretty simple. This specific style of scam began in the 1980s, when scammers from Nigeria would send official-looking letters overseas to potential marks overseas. Reportedly, these earliest letters often made the claim that the sender was a government official in the national petroleum department, or other government deparments. These letters claimed that said Nigerian government department had unspent budget funds, and the official wished to divert these funds abroad (for corrupt purposes). They simply needed some money to pay for customs fees, bribes to other officials, etc.
A possible reason that these letters may have seemed legitimate to Western marks is that in the late 1980s and early 1990s, Nigeria was ruled by the military dictatorships of Ibrahim Badamsi Babangida (1985-93) and Sani Abacha (1993-98). Babangida and Abacha were both notoriously corrupt, and credible claims have accused each of stealing billions of dollars from the Nigerian treasury and sending it overseas. It might be that the marks knew of this reputation for corruption, and were thus convinced that this opportunity to profit from corruption was genuine. On the other hand, as a general rule Americans are not terribly aware of news in Africa, so we should not necessarily presume that knowledge. However, for the part of the scammers, the fact that corruption was happening in their country probably provided some inspiration for the stories they told in their scams.
As time and technology progressed, the vector for this communication shifted from letters to email.
In Nigeria, there continue to be a class of customary chiefs/princes/emirs like the Emir of Kano, the Oba of Benin, the Shehu of Bornu. They continue to have prominent places in their community, and sometimes can mobilize that social prominence to lead political parties or serve in important positions in government. So, for a Nigerian scammer, making the claim "I am a Nigerian prince" would have a meaning and explain the source of wealth, in a way that Western marks probably would not understand. I'd wager most Americans do not realize that a "nigerian prince" is an actual thing, outside of email scams.
As you seem to indicate in your question, over the years this type of scam has become quite popular by scammers all around the world. Now, it is likely a majority of scammers claiming to be Nigerian Princes actually come from the United States, Europe, or other countries that are not Nigeria.