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. As we have seen, 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.
