Why did EVA Unit 01 only accept Shinji as its pilot?
EVA Unit 01 contains the soul of Yui Ikari, who disappeared during a contact experiment: in ep. 20, after Shinji achieves a high synchronisation rate and is absorbed into LCL in the EVA Unit, he comes into contact with his mother, and Gendo also makes a statement during one of these refusals that indicates a connection with Yui. Thus Unit 01's refusal to work with anyone or anything other than Shinji is linked to Yui's mother-child relationship with Shinji (a relationship she has with neither Rei nor an empty dummy plug) and Yui's own preferences.