There are several ways to secretly choose an opponent, including writing that player's name on a piece of paper that's kept with Stalking Leonin.

If you have multiple Stalking Leonins, you may choose a different opponent for each one. Be careful to keep track of which opponent is chosen for each Stalking Leonin.

If Stalking Leonin leaves the battlefield and returns, it's considered a new object. You may choose a new opponent and you may activate its last ability again.

If you didn't choose an opponent when Stalking Leonin entered the battlefield, you can't pay the cost to activate its last ability. This will usually happen because you gained control of another player's Stalking Leonin or because a creature already on the battlefield became a copy of Stalking Leonin.

If a creature is attacking a planeswalker you control, it's not attacking you. Stalking Leonin's last ability can't target it.