It is named after the city of Berkeley, California, where it was first produced--or, one might say, "discovered"--in 1949. Glenn Seaborg, Albert Ghiorso, and Stanley Thompson led the University of Berkeley team that prepared the tiny sample.
At the time, they were following a naming protocol by which the newly-discovered elements of the "actinide" group were named analogously to elements of the "lanthanide" group found in parallel positions in the periodic table of elements.
Since the new actinide, element 97, was parallel to Ytterbium--named after the city of Ytterby, in Sweden, where it was first found--the name "Berkelium" suggested itself, and was duly accepted.
By the way, Albert Ghiorso is credited with discovering more new elements than anyone else, and designed the first commercial Geiger counter.