Sakashima the Impostor has the following text (and more, but this is the relevant part):
You may have Sakashima, the Impostor enter the battlefield as a copy of any creature on the battlefield, except its name is Sakashima, the Impostor
Lazav, the Multifarious has somewhat similar text (also shortened to the relevant part):
{X}: Lazav, the Multifarious becomes a copy of target creature card in your graveyard with converted mana cost X, except its name is Lazav, the Multifarious
Between these two cards, it's possible to have Sakashima enter the battlefield as a copy of Lazav, and then use Lazav's ability. In most situations, text on a card which refers to that card by name is treated as "this card", so even though the creature in that circumstance is named "Sakashima the Impostor", activating its ability which says "Lazav, the Multifarious becomes a copy" will cause Sakashima to become a copy. However, I don't know whether the same applies to the part of the ability which affects the resulting name as well.
There are various other cards which would have similar effects when combined with Lazav, such as taking the ability directly with something like Necrotic Ooze that I assume would work the same way, but if not then feel free to limit consideration to just Sakashima.
In short, if a creature not already named Lazav, the Multifarious has and activates the ability of Lazav, the Multifarious, does its name change?