Of course the name of the city must be changed along with removing the crosses because after all, if the crosses offend the constitution, certainly the name does as well.
Also a little background into the local legends as to the origins of the name:
"The most popular theory is that sometime during the 1700s, a bishop, a priest, a Mexican army colonel, four trappers and four choirboys were attacked near the Rio Grande and only one choirboy survived. He put up crosses at the site and the area became known as El Pueblo del Jardin de Las Cruces, or City of the Garden of the Crosses."