Why when someone is sick do we say his mothers name not his father's?
The Zohar (Bereishis 84a, citing Psalms 86:16, "והושיעה לבן אמתך") says that it's better to pray using the most definite facts available. There can be a slight chance that the sick person isn't really the son of the man who is assumed to be his father (even though, for halachic purposes, we ignore this possibility and follow the majority - Chullin 11b), but his mother's identity is (almost) never in doubt.
I've also seen an answer that the mother is more associated with the body and the father with the soul. (The Zohar, Bereishis 79a, says this about Sarah and Avraham respectively; Ben Yehoyada to Berachos 55b says that this is generally true of all couples.) Thus, when we pray for a sick person's physical recovery, we invoke the mother's name; by contrast, when we pray for a deceased person's soul, we use the name of their father. (Though I believe there are variant customs on that last point too.)
The Ben Yehoyda (Brachos 55b s.v. Anna) answers the person needs merits, and since men have a harder time because they have Bittul Torah and other sins more specific to men, the mother's Zchus is better. The Birchei Yosef (Orach Chaim 284:1) uses this to answer why when someone has a father's Yahrtzeit and the other has a mother's the one who has Yahrtzeit for his father has precedence to say maftir. Basically, he says the same, that the father needs it more!!!
I heard in the name of Rav Heineman that the reason is because of סכנתא חמירא מאיסורא, and since we always have a ספק who the father is, it is better to use the mother's name to identify the person because by the mother there is no ספק. Even though this רוב of רוב בעילות אחר הבעל works very well in regards to איסור והיתר.
--but it could be that Rav Heineman was joking?
A source for using the name of the mother is the gemara Shabbos 66b:
Abayei said: Mother told me that all incantations should contain the name of the mother.
B'feirush davka Torah.
The source for mentioning the name of the individual's mother is King David's entreaty (Psalms 116:16): "Please, O Lord, for I am Your servant; I am Your servant the son of Your maidservant," wherein he specifies his mother (although his father was also a very righteous person).