Can Sakashima change its name to Lazav?
Yes, if a creature that is not named Lazav, the Multifarious activates the ability from Lazav, its name will become Lazav, the Multifarious.

Rule 706.9c says this:


  Some copy effects modify a characteristic as part of the copying process. The final value(s) for that characteristic becomes part of the copiable values for the copy.


In this case, Lazav's ability modifies the name as part of the copying process, setting it to "Lazav, the Multifarious", so the creature will have that name after the ability resolves no matter what name it had before.

With Sakashima in particular, after the ability from Lazav resolves, you will have two copy effects: one copying Lazav, the Multifarious, with the modifications made by Sakashima's ability, and one copying the target of the activated ability, with the modifications made by Lazav's ability. The second one has a later timestamp, so it overwrites the first one, and the resulting object just has the characteristics set by that second copy effect, including Lazav's name.

This would work differently if the ability said that it "doesn't copy" the creature's name, the way Vesuvan Doppelganger doesn't copy the creature's color.