I'm wondering if there are any linguistic reasons or rules that can be applied to find out which to use, or do I just need to memorize them individually?
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Well, all the Japanese counters have some irregularities. In this case, the standard counter for minutes is ふん, but in some numbers it changes.
1: いっぷん 2: にふん 3: さんぷん 4: よんぷん 5: ごふん 6: ろっぷん 7: ななふん 8: はっぷん 9: きゅうふん 10: じゅっぷん
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There is a rule when saying 'x past the hour': if x is 10, 20, 30, 40 or 50 you say '~じゅっぷん' not '~じゅうふん'.
For example 15:10 is じゅうごじじゅっぷん and 15:20 is じゅうごじにじゅっぷん whilst 15:15 is じゅうごじじゅうごふん.
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In fact, that's only one of the irregularities in this counter. 10 it's not the only irregular number of minutes. – Haury Mar 8 '17 at 22:42
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This answer only partially addresses the question. Perhaps you'd care to expand? – Halfway Dillitante Mar 9 '17 at 5:25
