Final answer:
Rahim Khan told Amir that Hassan was his half-brother, sharing the same father, Baba, which motivated Amir to return to Kabul.
Step-by-step explanation:
The secret Rahim Khan revealed to persuade Amir to return to Kabul was that Hassan, who was Amir's loyal friend and servant, was actually his half-brother. Rahim Khan disclosed that they shared the same father, Baba, which fundamentally changed Amir's perception of his past and instilled in him a sense of obligation to return and confront the realities of his family history in Afghanistan.