To save the Owlbear Cub in Baldur's Gate 3, follow these key steps:

1. Find the Owlbear Nest cave east of the Blighted Village. When you encounter the Owlbear Mother, do not kill the cub. You can interact using Animal Handling, Speak with Animals spell, or other skill checks to avoid conflict.
2. After leaving the cave, keep checking back until the mother is killed by goblins and the cub disappears from the cave.
3. The cub will then appear in the Goblin Camp, where it is held by goblins. Interact with the cub and succeed in a skill check like Animal Handling or Persuasion to let it recognize you.
4. You can then attempt to free the cub via different options including playing or declining a chicken-chasing game, intimidating or persuading goblin leader Krolla, or buying the cub.
5. After freeing the cub, it will eventually follow your scent back to your camp after several long rests.
6. When the cub appears at your camp, feed it on the first night and heal its injuries on the second night to ensure it stays permanently as a pet.

Additional tips:

• You can let the mother live by knocking her out and healing her, which prevents goblins from finishing her off.
• The cub must not be attacked during any combat.
• The cub can be recruited as a camp follower once it joins the camp.

This process is a bit tricky but rewarding, allowing the Owlbear Cub to permanently join you as a companion in the game.