This was once two similarly-named streets. La Veta Place was laid out in 1886 on the Angeleño Heights tract, between Bellevue Avenue and Kensington Road (then more accurately named “The Crescent”, owing to its shape). Thirty years later, Laveta Terrace was introduced on the other side of Sunset Boulevard. The old joined the new in 1912 by city ordinance. My educated guess is that the name comes from la veta, Spanish for “the mineral vein”. In 1880s Los Angeles, many capitalists investing in local real estate also had mining interests outside the city; furthermore, a Colorado town was christened La Veta in the 1870s and a major thoroughfare in the city of Orange was called La Veta Avenue in 1887, so there was obviously a trend going on. Yet the men behind Angeleño Heights (see Wallace Avenue and Everett Street) had no known mining background nor any connection to a La Veta or a Laveta. The mystery remains.
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