El Paso has been nicknamed the “Sun City” because it gets over 300 days of sun each year. Many Texas cities have limited rainfall, so the nickname shouldn’t be exclusive to El Paso. Arizona also has a “Sun City,” founded in 1960.

“Sun City” became attached to El Paso also by 1960.


Wikipedia: El Paso, Texas
The sun shines 302 days per year on average in El Paso, 83 percent of daylight hours, according to the El Paso Weather Bureau. It is from this that the city is nicknamed, The Sun City. 

28 April 1960, Deming (NM) Headlight, pg. 3:
EL PASO, TEX.—Deming Realty Co. of Deming has been named agent in Deming area for Sun City, a mile-high 207 square mile ranchland area scheduled for conversion into a residential and industrial metropolis.

The large land development was touched off by the purchase of 107,000 acres of adjacent land by Arthur Rubloff of Chicago. Sun City lies 21 miles east of El Paso.
(...)
The historic Butterfield Trail runs through the Sun City area, which has 362 days of sunshine a year.

4 September 1960, New York Times, pg. 204:
Walls in the offices of Mount Franklin Homes, Inc., and Sun City, Inc., El Paso’s two leading land offices, which are run by the O’Leary Brothers, contain huge maps of the Horizon City project, with the proximity of the two land speculation developments clearly indicated.

10 December 1960, Chicago Daily Tribune, pg. 12:
A record breaking snowstorm isolated the “Sun City” of El Paso, Tex., yesterday and paralyzed travel over a three state area.