Can't boot Apple Hardware Test with correct install disk that has AHT
My first thought was that perhaps you had somehow updated the firmware and were trying to use an AHT disk that couldn't boot due to this. This is a possibility because your Mac Mini came pre-shipped with Mac OS X 10.5.6, but the firmware update (which you do have installed) requires Mac OS X 10.5.7. However, then I realised your question says you only just ran the AHT last week. So, unless you somehow updated the firmware in the past week, this isn't likely to be your problem.

Since you can't seem to get your disk to work, the only other option I can think of is to download AHT again and run it from a USB. The steps to do this are as follows:


Create a bootable USB flash drive by using Disk Utility's Erase function. Make sure you choose OS X Extended for the format and the GUID Partition Map for the scheme. For the purposes of these steps, also give your USB the name AHT when you're erasing it.
Download the AHT for your particular Mac Mini here
Mount the downloaded image (it should mount as AHTCThree)
Now you will need to copy the AHT to your USB flash drive. To start, Launch the Terminal app (usually found within the Utilities folder)
In the Terminal window, enter the following:

cp -r /Volumes/AHTCThree/System /Volumes/AHT/
Now we need to make the USB flash drive bootable, so enter the following line in Terminal:

sudo bless --folder /Volumes/AHT/ --file /Volumes/AHT/System/Library/CoreServices/.diagnostics/diags.efi --label AHT
Enter your Admin password
Leave the USB flash drive plugged in and restart the Mac Mini
Immediately hold down the Option key
Select the AHT from the list and boot up


NOTE for other readers: - These Terminal commands were written for this specific question. However, these can be adapted for your use by replacing the AHTCThree with whatever name your downloaded disk image mounted as at Step 3. For example, if at step 3 your downloaded disk image mounted as AHTEOne, then at Step 5 you would use AHTEOne instead of AHTCThree within the Terminal command.