Of all the ways I could come up with to pronounce the word "busy", "bizzy" would be very low on my list. At least "bussy" or "boosy". Why "bizzy"?
The problem isn't so much that "busy" is pronounced "bizzy", but that "bizzy" is still spelled "busy". An awful lot of English spelling is based on a language that would sound very foreign to most English speakers today -- English as it was before the Great Vowel Shift (and at a point when we still had velar fricatives, which is why there are words spelled with "gh"). The language has undergone major changes in pronunciation, while the spelling has merely been tidied up a bit so that words are spelled the same way every time. (Mostly.)
If you really think about it, almost none of English spelling makes any sense with regard to vowels. It used to, at least to the same extent that alphabetic writing makes sense in any language. Your example "bussy" would have been pronounced roughly like the Modern English word "boozey", but with a doubled/stressed "s"; "boosy" would have had a long "o" (both in the modern schoolroom sense that "the vowel says its name", and in the sense of duration).
What happened during the Great Vowel Shift was systematic. That's why we're able to derive spelling rules dealing with things like "silent Es" and so forth -- the "real" orthography applies to a differently-pronounced language that has changed in a particular way, so if we know how it has changed, we know how to apply the rules of the old language to the new one. But there are also differences that are dialectical -- the accepted spelling of some words reflects a dialect of Middle or Early Modern English that may have won the war, but it lost some of the battles. If "busy" is spelled "busy" today, it's probably because it was pronounced that way in the London dialect at one time. Unfortunately, written documents are the only recordings we have of the pronunciation, so we can never know exactly how it was pronounced, but we do know that there was at least some method to the madness back then.
As Ham and Bacon points out, "busy" is an old word. The problem with giving Old English roots (in this case, "bisig"), though, is that Old English was itself a bundle of dialects. Generally, what's reflected in etymologies is a "majority rules" case. That does not imply that the Old English dialect that gave rise to the Middle English dialect that made "busy" (or "busie") seem like a good idea did not also use a "u" or "y" (high front rounded) sound in their version of the word.
The spelling of busy (and bury) is the result of dialect mixture. Different Middle English varieties had different outcomes of Old English short /y/. In the East Midlands variety that underlies the standard, it became short /u/ as in blush; in Kent, short /ɛ/ as in merry (for people who pronounce it with the same vowel as in met, anyhow); in the West Midlands, short /i/ as in bridge: all three of these words had short /y/ in Old English (blyscan, myrige, brycge). Busy is an East Midlands spelling for an West Midlands pronunciation; bury is likewise a East Midlands spelling, but for a Kentish pronunciation.
UPDATE: I'm dyslexic about left and right, which also extends to east and west. Obviously it's East Midlands (where London is) that underlies the standard, so that's what this now says.
I reckon its in its etymology. It originally came from the Dutch word
O.E. bisig "careful, anxious, busy, occupied," cognate with O.Du. bezich,
The root was actually spelt with a 'z', and it was changed to 's' later. This can also be seen in the word "organisation":
1375–1425; late Middle English organizacion < Medieval Latin organizātiōn
Note that its root, Latin, was spelt with a 'z'.