The mini-tower should have at a minimum:
- dual or quad core, 8 GiB RAM max (4 slots)
- optical and 3.5" hard drive
- one free drive bay for either 2nd optical or 2nd hard drive
- integrated graphics *AND* PCIe x16 graphics slot with PCIe power connector
- two open PCIe x1 slots (TV tuners, major sound card, ...)
How could that computer possibly be called a Mac mini?