Yes! Precious is a legitimate name. Its usage goes all the way back to early Medieval England, and it was quite a common female name in its myriad forms across Western Europe.
It is first attested as a female given name in England in the year 1203, then again in 1279 and in 1327. It was common in Medieval Europe to give females names with nice attributes.
Preziosa appears in Medieval Italy and Preciosa is occassionally used among Spanish-speakers, sometime in reference to the Precious Blood of Christ.
Precious seems to have fallen out of usage by the 15th-century and was not revived until the late 20th-century. In the United States, it is more prevalent in the African-American community.
It is currently 958th most popular female name in the United States (2010).