When praying for a person in a dire predicament, such as an ill individual, his or her Hebrew name is mentioned along with his/her mother's name. For example, "Avraham ben (the son of) Devorah" or "Rachel bat (the daughter of) Chaya." The same holds true when requesting a blessing from a tzaddik: it is customary to write in the note the names and mother's names of all those on whose behalf the petitioner is requesting a blessing or prayer.
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).
King David says: "Bring salvation to the son of Your maidservant," We can always be sure who a person's mother is, whereas the father's identity is never absolutely unquestionable. When praying for someone's life, we obviously do not want to confuse his/her identity through using mistaken lineage.
On a deeper level, a Jew's spiritual essence is inherited via his or her mother, as evidenced by the fact that Judaism is passed down matrillineally. When praying for another, we want to emphasize their essential and eternal link to G‑d, as derived from their mother's side.
Ideally, Jewish names should be used for prayer. If, however, only the mother's non-Jewish name is known, it should be mentioned, as in "…ben (or bat) Elizabeth." In a case when the person's mother's name is entirely unknown, the ill person's name should be appended with "…ben (or bat) Sarah," referring to the matriarch Sarah, who is the mother of all Jewish people.
When praying for a non-Jew, we mention the person's name along with his/her father's name.
May we only need pray for continued good health for us all!
