Question: Why is it that when a person is called up to the Torah, we mention his father's name, while when praying for the recovery of someone who is sick we mention the patient's mother's name?

Answer: In general - and especially according to Ashkenazi custom - a person is known as "So-and-so, son (daughter) of (father's name)" because this is the way in which people used to be identified - according to the name of the father. 
Since the prophet describes "Rachel weeping over her children" (Yirmiyahu 36), and according to the Gemara it is specifically Rachel's prayers that are answered by God with the promise that "the children will return to their borders", therefore we make 
mention of the mother's name in situations of danger or trouble, such as when praying for a person's recovery from illness. 
I believe that it does no harm - in fact, it is beneficial - to mention the mother's name in documents such as the ketuba, which is read as part of the wedding ceremony, as well as on a gravestone. A mother is certainly an important figure (to say the least) in 
her child's life. Among oriental communities, the mother's name is mentioned far more often, and especially in the "hashkava" prayer that is recited during shiva.