Why was the sign given to convince the Jewish believers at Cornelius' house?
The sign was given to convince the Jewish believers at Cornelius' house that God was now saving Gentiles without requiring them to become proselytes of Israel. Since those Jews would return to Jerusalem, the sign confirmed they should spread the word that God was turning to Gentiles independently of Israel, on His own ground, not through the nation.
Les explains that the sign was particularly used to convince these Jews because they were going to leave Cornelius' house and go right back to Jerusalem. They should have simply spread the word that God was now ready to turn to the Gentiles without Israel. The sign thus confirmed to the Jewish believers that a Gentile could be saved without becoming a proselyte of Israel. It showed that God was now dealing with the Gentiles on His own ground, not on the basis of using the nation of Israel. Les notes that they did not spread that word; Acts 15 indicates Peter and those who evidently went with him had never said a word for another eight or ten years among the children of Israel about this change.