The word translated as saint in scripture is the Greek word "hagios" which means “consecrated to God, holy, sacred, pious." Some translations rather than translating the term as 'saints', instead translate it as Holy Ones. The idea of the word “saint†is that a person has been set apart for the Lord and His kingdom. So a saint according to scripture is a person who has been sanctified (set apart for use) by the Holy Spirit of the Lord having been cleansed by the blood of Jesus. Here are some scriptural references:
2Th 2:13, 1Pe 1:2, Eph 5:26, Rom 12:1-2
It seems that the scriptures indicate that one became a saint when one received the gift of God's Holy Breath (Spirit) which acted as a "seal" of sorts marking those as sanctiified. (see Eph. 1:13 ; 2Cor 1:21 & Rev 9:4)