Can I activate Multani, Yavimaya's Avatar's ability in response to itself?
You can activate Mutani's ability in response to itself.
Your opponent was wrong in stating that activating it caused Multani to leave the graveyard. Activated abilities have a cost and an effect, separated by a colon.

602.1. Activated abilities have a cost and an effect. They are written as &quot;[Cost]: [Effect.] (...)&quot;
602.2. To activate an ability is to put it onto the stack and pay its costs, so that it will eventually resolve and have its effect.

Returning Multani to your hand is part of the effect, so it will only happen on resolution. Because you can respond to your own abilities by retaining priority, until that ability resolves, you may activate the same ability again as many times as you're able to pay the cost (returning two lands to your hand).

116.1b A player may activate an activated ability any time they have priority.
116.3c If a player has priority when they cast a spell, activate an ability, or take a special action, that player receives priority afterward.