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According to Wiktionary, hypocrisy (/hɪˈpɑkɹəsi/) and hypothesis (/haɪˈpɒθɪ̈sɪs/) are both coming from French, from Latin, and earlier from Greek, with the same root ὑπό (hupó).
However, the ‘y’ in hypo- is pronounced completely different in both cases. One plausible theory could be that hypocrisy used to be written ipocrisie in the XII century, and the pronunciation could have diverged at that time.
What rule to use to pronounce words starting with hypo-?
Yes, the use of /ɪ/ in hypocrisy (and the related hypocrite and hypocritical) is probably related to the derivation from French. It is an older pronunciation pattern. The Oxford English Dictionary says:
The first vowel in Greek ὑπο-, Latin hypo-, is short, and all the early words in English were introduced with the y short, as in hypocrite, hypocrisy, etc. The y is marked as short in all compounds with hypo- in Pronouncing Dictionaries down to the middle of the 19th c. Some later Dictionaries, while retaining short y under stress, primary or secondary, as in hypocaust, hypothetic, make it long /aɪ/ in unaccented syllables, as in hypothesis, hypotenuse. But the later tendency in the South of England has been to treat y in all positions except before two consonants as /aɪ/, and, against etymology and history, to say hȳposulphate, hȳpostatical, etc.
I'm not actually sure which other words the "etc." here refers to: I couldn't find any other hypo- word that is pronounced with /ɪ/ instead of /aɪ/ in present-day English.
One general pattern in the pronunciation of polysyllabic English words is that a single vowel letter other than u (so, any of a e i o) in a stressed syllable usually corresponds to a "short" vowel sound if the vowel is followed by at least one consonant sound and the stressed syllable is followed by at least two other syllables. This tendency has been given various names, like Luick's law and trisyllabic laxing.
However, there are exceptions, among which are many compound or prefixed words of Greek origin. The use of such compounds is I believe a relatively "recent" phenomenon, in a certain sense: although the word hypocrisy technically originates from a Greek compound/prefixed word, a native English speaker won't necessarily feel like it is a compound word, while a word like hypoglyc(a)emia is more transparently a compound.
For some reason, the sound /aɪ/ is quite common in this context, occuring in combining forms like chiro- cymo- cyto- glyco- gyro- hydro- hypo- iso- kymo- lyso- micro- mito- nitro- phylo- psycho- pyro- rhizo- Sino- spiro- stylo- xylo- zymo-.
For example, the pronunciation of the first syllable of the word hypothetical violates the "trisyllabic" rule.
I'm not sure why or how this exception came to be. Perhaps, as the combining forms ending in -o came to be used more freely to make compounds in English, people came to think of them as being more like words of their own, and so people don't intuitively feel like the first syllable is followed by more than one syllable. In that case, the pronunciation would be similar to the regular pronunciations with long vowels of a number of independent words ending in o like veto, halo, hero, zero. But that wouldn't explain why the exceptionally-pronounced combining forms seem to primarily consist of forms with i or y pronounced as /aɪ/.
The same tendency exists to some extent in -o combining forms with certain other vowel letters, but to a lesser degree. As far as I can tell, there are no or almost no examples of a being pronounced as a "long vowel" in this position (only maybe strato- and nano-, compared to acro- agro- caco- macro- phago- scapho- stato- all pronounced with "short a"). But the letters e and o have pretty unpredictably variable vowel length in this context, with a number of combining forms having multiple pronunciations (such as chrono- sono- phono- homo- tropo- topo- and geno- meso- telo- xeno-). I wrote an answer a while back about the pronunciation of xeno- that talks about some of the -e_o- combining forms: Pronunciation of “xeno-”
Perhaps the "trisyllabic laxing" pattern is mostly a historical relic at this point, and so people don't feel any strong pressure to use pronunciations that conform with the pattern. But that doesn't do much to explain why people don't use pronunciations with /ɪ/ for the above-mentioned forms (there are some combining -o forms that are pronounced with /ɪ/ rather than /aɪ/, mostly ones with multiple syllables that end in -ino-, like carcino- fibrino- hallucino- pepsino- peptino- plasmino- trypsino- urobilino-, but also a few miscellaneous forms like mytho- litho- chryso- philo- miso- erythro- and one pronunciation of cyclo-).
The pronunciation of i or y in unstressed syllables is a separate, but related issue. There is a lot of variation, so in fact I don't even know of any general rule that can be used to determine if /aɪ/ is expected (the rules I remember being given in Walker's Critical Pronunciation Dictionary are pretty long and mention some words with multiple pronunciations).
In the case of hypocrisy and hypothesis, however, it seems reasonable to assume that the pronunciation has simply been influenced by the related words with different stress patterns.
There is no rule. Most native speakers of English have no knowledge of French, Latin or Greek. They pronounce words as they hear them from others or, if they have never heard them spoken, how they judge those letters ought to sound.
Hypocrisy is encountered much more frequently in daily life than hypothesis, which is a rather academic technical term. It is not surprising that the pronunciation of the two words differ.
When the so-called mad cow disease appeared in the UK, it was known technically as bovine spongiform encephalopathy, with the first syllable definitely pronounced as "enk...". If, at that time, you had needed to have your head examined you would have had an encephalogram, in which the first syllable would almost certainly have been pronounce "ens..."
