is there any person after whom this area is named?
James Gadsden
Gadsden Purchase

The first draft was signed on December 30, 1853, by James Gadsden, U.S. ambassador to Mexico, and by Antonio López de Santa Anna, president of Mexico. 
who negotiated the purchase of the area south of gila river UNANSWERABLE what do you mean by this purchase? It is a region of present-day southern Arizona and southwestern New Mexico.
wiki:1135016
Gadsden Purchase [SEP] Introduction
After the devastating loss of Mexican territory to the U.S. in the Mexican–American War (1846–48) and the continued filibustering made by New Mexico governor William Carr Lane in the zone, Santa Anna may have calculated it was better to yield territory by treaty and receive payment rather than have the territory simply seized by the U.S.