Q: What are your thoughts about why the people thought Jesus, while on the cross, was calling out to Elijah? Was it for rescue?

A: As recorded in Mark 15:33-36, when Jesus was on the cross he cried out in Aramaic "Eloi Eloi, lama sabachthani?" which means "My God, My God, Why have you forsaken me?" (compare Psalm 22)

Some of the people standing by mistook the word "Eloi" ("My God") for the name Elijah.

According to Malachi 4:5 (Malachi 3:24 in the Hebrew Bible), the LORD was to send Elijah before the "Great and Terrible Day of the LORD," so Elijah came to be associated with deliverance from that calamity. This return of Elijah has also long been associated with Passover. Perhaps that is why the people in the crowd thought Jesus was calling for Elijah to come and save him.