The traditional interpretation of the Hebrew form of the name "Nazareth" is generally understood to be נֵצֶר (netser), which means "sprout" or "shoot." This interpretation is based on its usage in messianic prophecies in the Hebrew Bible, such as Isaiah 11:1. cf Strongs #5342.
"And he has gone out, a rod from the Stump of Jesse, a branch [netser] from out of the roots of himself is bearing fruit." (Isaiah 11:1 RBT)
However the root verb נצר means to watch over/protect as נֹצְרָה (notzrah) meaning "protectress" or "guard" and נְצֹרֶת (netsoret) meaning "watch-tower." (See Strong's #5341)