This has to do with English stress patterns and prosody. English stress patterns are enormously complex and have many, many, many exceptions.
'Bicycle' has a short vowel in its second syllable because the prefix bi- is a stress-bearing prefix and can take primary stress (prominence). So when the primary stress moves from the cy to bi-, the cy gets unstressed and the diphthong [aɪ] gets shortened to [ɪ].
The diphthong in the second syllable of 'recycle' doesn't get reduced because the prefix re- doesn't often take primary stress, so the primary stress remains on the first syllable of -cycle.
There are two main kinds of syllables on the basis of loudness; stressed and unstressed. A stressed syllable is the prominent/loudest syllable in a word. Not all syllables in English words receive the same level of loudness (i.e. all syllables in a single word cannot have primary stress).
There are three levels of stress in English:
- primary stressed syllable: the loudest syllable in a word
- unstressed syllables: syllables that don't have stress at all
- secondary stress: syllables that aren't unstressed, but aren't as loud as primary-stressed syllables either (lie in between the two)
In the IPA, primary stress is represented by a preceding superscript vertical line [ˈ] while secondary stress by a preceding subscript vertical line [ˌ]. So the word pronunciation can be transcribed as [prəˌnʌnsiˈjeɪʃn]: the second syllable has secondary stress and the second last syllabl has primary stress.
The word 'cycle' is stressed on the first syllable and is pronounced /ˈsaɪkəl/ (with the diphthong [aɪ]).
The diphthong [aɪ] (as in bite) rarely occurs in an unstressed syllable. It has a systematic relationship with the short vowel [ɪ] (as in kit). This relationship is reflected in many words such as f[aɪ]nite - inf[ɪ]nite and many other remnants of Trisyllabic Laxing; div[aɪ]ne-div[ɪ]nity, der[aɪ]ve-der[ɪ]vative, and in dr[aɪ]ve-dr[ɪ]ven etc.
From this relationship, we can infer a general rule of thumb that [aɪ] will only occur in a stressed syllable and when that syllable gets unstressed, [aɪ] will get reduced to [ɪ].
The reason why 'bicycle' is pronounced /ˈbaɪsɪkəl/ is that the prefix bi- is a stress-bearing prefix and can take primary stress. Also note that English has a strong tendency towards antepenultimate stress. It means that English words tend to be stressed on the antepenult (third-last syllable).
'Cycle' has primary stress on its first syllable and when the prefix bi- is prepended to it, the primary stress moves from cy to bi-, leaving the cy unstressed. And as I said above, when a syllable having [aɪ] gets unstressed, the diphthong [aɪ] usually reduces to the short vowel [ɪ]. That's what happens in bicycle.
Re- on the other hand does not often take primary stress1, so when it's prepended to cycle, the primary stress remains on the cy. That's why the re- remains unstressed and the cy becomes the prominent syllable.
However, the cycle in unicycle, quadricycle, motorcycle is pronounced with the diphthong [aɪ]. It has been explained in this answer on ELU [Modified]:
motorcycle, quadricycle and unicycle consist of a pair of trochees: MO-tor-CY-cle, QUAD-ri-CY-cle, U-ni-CY-cle. The primary stress in these words falls on the first syllable, but the secondary stress falls on the first syllable of -cycle. In this context the vowel retains its full pronunciation i.e. it still receives stress, so the diphthong doesn't get reduced.
- Re- can take primary stress in some words, as Colin Fine and Canadian Yankee pointed out in the comments below. It seems to me that disyllabic nouns that have the prefix re- have primary stress on the re-, whereas disyllabic verbs don't have primary stress on the re- (cf. REcord and reCORD). I think trisyllabic nouns having the prefix re- follow the antepenult stress rule, so they get primary stress on the re-.