I installed snow leopard on my mac mini and now it takes roughly 10 minutes to boot

Mac OS X 10.6: If you see a blue screen at startup.

You may need to reinstall Snow Leopard:

Reinstall OS X without erasing the drive

Do the following:

1. Repair the Hard Drive and Permissions

Boot from your Snow Leopard Installer disc. After the installer loads select your language and click on the Continue button. When the menu bar appears select Disk Utility from the Utilities menu. After DU loads select your hard drive entry (mfgr.'s ID and drive size) from the the left side list. In the DU status area you will see an entry for the S.M.A.R.T. status of the hard drive. If it does not say "Verified" then the hard drive is failing or failed. (SMART status is not reported on external Firewire or USB drives.) If the drive is "Verified" then select your OS X volume from the list on the left (sub-entry below the drive entry,) click on the First Aid tab, then click on the Repair Disk button. If DU reports any errors that have been fixed, then re-run Repair Disk until no errors are reported. If no errors are reported click on the Repair Permissions button. Wait until the operation completes, then quit DU and return to the installer.

If DU reports errors it cannot fix, then you will need Disk Warrior and/or Tech Tool Pro to repair the drive. If you don't have either of them or if neither of them can fix the drive, then you will need to reformat the drive and reinstall OS X.

2. Reinstall Snow Leopard

If the drive is OK then quit DU and return to the installer. Proceed with reinstalling OS X. Note that the Snow Leopard installer will not erase your drive or disturb your files. After installing a fresh copy of OS X the installer will move your Home folder, third-party applications, support items, and network preferences into the newly installed system.

Download and install the Mac OS X 10.6.8 Update Combo v1.1.

Try booting in Verbose Mode, Cmd-v at startup. This will bring you to a screen with text showing the boot sequence. Write down the messages or take a picture when it hangs. Also, have a look in Console>system.log in Utilities for any relevant messages around the time of bootup.

You don't exit Verbose Mode, it will exit by itself and continue booting normally when it's finished the boot sequence. There is nothing to do but wait. Yes, where it hangs means where it appears to stop for a while.

This might give a clue about why booting is taking so long.

You might also see if applying or reapplying the 10.6.8 Combo Update fixes this.

http://support.apple.com/kb/DL1399

I like to apply updates from Safe Boot. Hold down the Shift key at startup. Give it longer to boot than normal, which, in this case is already abnormal. But, who know, a Safe Boot might actually boot faster and give a clue about the long "normal" bootup.

The Safe Boot might clear the blue screen issue. Do a Safe Boot, keep everything closed, leave it there maybe 3-4 minutes doing nothing, then boot again normally.

Also, check to see that the normal boot volume is selected in System Preferences>Startup Disk.

Are you getting the spinning beachball, or a rectangular progresss bar? But, anyway, give it time, Safe Boot is supposed to take much longer than a regular boot. Among other things, it's doing a directory check and repair, if necessary.

What happened after you restarted normally?

If you go to Disk Utility and select the drive, the top-most listing, not the volume indented below it, does it show SMART Status Verified?

Also might try getting the free demo of SMART Utility to check the health of the drive.

http://www.volitans-software.com/smart_utility.php

Can you boot again to Safe Boot, open up Disk Utility and Verfiy Disk and Repair Permissions for the volume.