Why is "bicycle" pronounced differently from other obviously related words?
Although such variation could be the result of things like when the word was borrowed into the language, this variation is probably due to the prosodic structure of the words; we get different vowel sounds because of the way that stress influences vowel quality in English.
In English, unstressed vowels are generally reduced. Take the word record for example.
The noun form takes first syllable stress: [ˈrɛ kɚd]. If you aren't familiar with IPA, note in particular the [ɛ] vowel and the [ɚ] r-colored schwa vowel. Schwa (and sometimes [ɪ]) is what often shows up in reduced, unstressed vowels in English. Since stress is on the first syllable, we get the r-colored schwa in the second syllable.
The verb form takes second syllable stress: [rə ˈkoɹd]. Note that the now-unstressed [ɛ] was reduced to [ə], schwa, while the [ɚ] now retains the full [o] sound (with an [ɹ] "r" after it).
Now for bicycle and unicycle:
bicycle [ˈbaɪ sɪ kəl] has primary stress on the first syllable. The "cy" syllable is unstressed, and so it is pronounced as a mere [sɪ] (or even [sə] by some). Note that tricycle has the same stress pattern, and has the same vowel.
unicycle [ˈju nɪ ˌsaɪ kəl], being a longer word, has multiple stressed syllables. Primary stress, again, falls on syllable #1, but the important thing is that secondary stress falls on syllable #3, the "cy" syllable. This means that "cy" is pronounced as a full [saɪ] instead of a reduced [sɪ]. Note that motorcycle has the same stress pattern and, again, same vowel.
The reason boils down to English prosody and stress patterns.
First, a general observation that the vowel [aɪ], which is the first vowel in cycle, very rarely occurs in unstressed syllables. Unstressed syllables that would otherwise contain this vowel tend to reduce it to [ɪ] or even [ə], as nearly all English vowels are so reduced when they occur in unstressed syllables. From this we can infer a general pattern that the vowel in cycle will only be pronounced as [aɪ] as long as the vowel is stressed.
The second observation is that both motorcycle and unicycle consist of a pair of trochees: MO-tor-CY-cle, U-ni-CY-cle. The primary stress in both 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.
Bicycle, however, has only one syllable before the -cycle, and the primary stress falls on this syllable. English does not allow two stressed syllables in a row in native words, which means that the second syllable of bicycle has no secondary stress. In turn, this means that the vowel [aɪ] is reduced to just [ɪ] in this context.
There are a few exceptions to this rule about two stressed syllables in a row, but they are all compounds or neologisms: fly-by, cry-baby. And in fact, the oldest spelling of the word bicycle is bi-cycle, where the hyphen is meant to indicate that the word is a compound and suggests the pronunciation b[aɪ]-c[aɪ]cle. But as the word bicycle became more common, the word was adapted to the native stress pattern, requiring reduction of the vowel in the second syllable.