What was the thought process of the family that led them to visit Shirdi for their son's ailment?

The family's journey to Shirdi was a step taken out of desperation and faith after all else had failed. Chapter 34 explains that after exhausting all medical treatments for their son's bone-cyst, the relatives first suggested worshipping deities. When that did not seem to bring relief, word reached them of a famous saint, or 'Avaliya', in Shirdi. They heard that Sai Maharaj could heal diseases just by his sight and that his Udi had miraculous properties. Seeing this as a last resort, the parents decided to try it, thinking, "What is there to lose in gaining this experience?" They hoped that by bowing at his feet, this final measure might avert the calamity befalling their son.