Question: 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"?

Answer: 
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 [`bezig`:](http://www.etymonline.com/index.php?search=busy&searchmode=none)

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> O.E. bisig "careful, anxious, busy, occupied," cognate with O.Du. bezich,
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The root was actually spelt with a 'z', and it was changed to 's' later. This can also be seen in the word ["organisation":](http://dictionary.reference.com/browse/organisation)

> 
> 1375–1425; late Middle English organizacion < Medieval Latin organizātiōn
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Note that its root, Latin, was spelt with a 'z'.