Why did Jesus use mud and rub it on the blind man's eyes?

It was a way of anointing the man's eyes which were then healed miraculously.

AnswerMark 6:5-6 states that, in his own country, Jesus could do no mighty work except heal a few sick people. He marvelled at their unbelief. Read literally, this means that Jesus was powerless to perform miracles unless the people believed. Because of this need for people seeking cures to have faith, it was customary in ancient times for holy men to use spit and other tricks when curing afflictions.

Two early miracles are keys to establishing the authenticity of all miraculous cures said to have been performed by Jesus. They are the cure of a deaf and dumb man (7:32) and of blind man (8:23). In these, Jesus had to use spit and even had difficulty in completing one cure successfully. The manner of performing these two cures was so orthodox that Mark is likely to have created the stories simply based on his knowledge of anecdotes about holy men in general. The only practical alternative is that Jesus had learnt magic tricks from others.

In later passages, Mark becomes more confident of Jesus' powers and not only is spit no longer needed, but the miracles become progressively more awe-inspiring, a trend reinforced in the later gospels.