David Horner, professor of biblical and theological studies at Biola University, seeks to answer the age-old dispute:
[W]hile consideration of the Latin original of Augustine’s name does not determine a single, grammatically obligatory English pronunciation, it does suggest that aw-GUS-tin is the more fitting or appropriate pronunciation. This is because the latter most closely preserves the distinctive placement of the accent in the original.
. . . Augustine’s Latin name is properly pronounced ow-goost-EE-nus, with the accent on the penultimate syllable. The pronunciation of aw-GUS-tin preserves that accent pattern: when the final syllable is dropped from the Latin name in forming the anglicized name, aw-GUS-tin retains the accent on the penult rather than wrenchingly shifting it to the antepenult, as in the case of AW-gus-teen. In this way aw-GUS-tin is closer to the original pronunciation pattern, and it thus constitutes a more natural and appropriate pronunciation.
For this reason, the Oxford Encyclopedic English Dictionary, recognized universally as authoritative in things most fine and fitting, lists aw-GUS-tin as the single recommended pronunciation.
You can read the whole thing here, humorously written in the style of a medieval disputation in response to a piece in a similar vein written by his colleague Garry DeWeese.