I haven't found a definitive answer, although Harvey H Wilcox filed a deed naming "Hollywood, California" in 1887, some 36 years before the "Hollywoodland" sign was erected, so the name "Hollywood" would have already been in common usage by 1923. I guess the sign was put up in the most prominent location above the new real estate development. From what I've read, Mount Lee was an "unnamed peak" (
Wikipedia) until the transmitter was built. To add to the confusion, the book
'Hollywood, 1940-2008' by
Marc Wanamaker has the following misleading caption.
books.google.com
Some further reading on Mount Hollywood:
Small Mountain, Big Dreams
A History Of Griffith Observatory