How should one parse “Southern Poverty Law Center”?
[ol]
[li]{Southern Poverty} {Law Center}: A law centre dedicated to the study of southern poverty.[/li][li]Southern {Poverty Law} Center: A centre, located in the south, which studies poverty law.[/li][li]{Southern Poverty Law} Center: A centre which studies the poverty law of the south.[/li][/ol]
The SPLC started as a civil rights law firm. It’s not clear from its website why that particular name was chosen. However it seems logical that your number two is closest. A center, located in the south, which practices law. Poverty law may have been an euphemism for civil rights law back in 1971.
If the wiki is to be believed, it’s “purpose is to combat racism and promote civil rights through research, education and litigation.” So it’s a law center.Southern Poverty Law Center - Wikipedia
http://www.learningtogive.org/papers/index.asp?bpid=160 So it specializes in poverty law.
And it’s in Montgomery, Alabama, which is in the South. Pretty close to #2.
At least it’s not a one-eyed one-horned flying purple poverty center.
When you found a law center about southern poverty or southwestern wealthiness, I’ll be pleased to let you name it whatever you like. 
Am I the only one who listens to the lyrics?