Evidence seems to suggest that elephants do not like small animals running frantically around their feet particularly as they cannot see them properly. Naturally in the wild quick moving small animals like snakes are a danger and so elephants would try to get away from them as quickly as they could. This may be the reason that elephants appear to be afraid of mice.
Fable featuring a mouse and an elephant
An ancient fable tells the story of a duel between an elephant and a mouse for the position of king among the animals. The elephant sat and laughed at the mouse until the mouse climbed into his ear (or in another story his trunk) and nipped him hard and drove him mad until the elephant ultimately admitted defeat.
I like my elephant and mouse to be friends, so here they are.