Yes, you can
Yes, you can upgrade the core duo or core solo processors and install Lion OS (10.7). Go slow because it's tricky getting the heat sink off the processor and back on with the four plastic posts and springs holding down the heat sink. It's easy to lose springs. The plastic barbs break off easily so you will want to get plastic screws and bolts ahead of time to replace the brittle posts that you break (I broke two of the four). The heat sink is also "welded" on to the old processor so be patient while you persuade the two to break apart. Also, the old thermal paste has to be worked off the new processor and heat sink a number of times using ArticClear remover and purifier. You can then add fresh thermal paste to the replacement processor (thin layer).
Heat Sink
Check the orientation of the heat sink also because the new processor is not oriented the same way as the original and if you use the outline of the old processor on the heat sink as a guide you will have to take the heat sink back off and turn it 90 degrees so that the fins are oriented correctly. Also, if you removed the heat sensor wire so that you can clean the heat sink don't forget to plug it back in before you lock down the heat sink again.
Core 2 Duo processor
Finally the new processor does not always drop easily into the ZIF (Zero insertion force) socket as you might expect. In my case a little pressure was needed probably because some of the pins were slightly off from the time it was removed from its previous home. I installed a T7600 and 4GB of memory into a Core Duo A1176 (late 2006) Mac Mini running OS 10.6.8 which I installed before upgrading the processor.
Warning
The mini will not boot again until you remove one of the memory sticks, leaving 2GB. This is because you will need to update to Lion before updating the bios to 2.1 so that the extra memory will be recognized. It will see 3GB of the 4GB you install. At current prices it's worth it. It's a scary moment when the Mini refuses to boot up and such a great sound after you remove one of the sticks and try again.
I installed a new HD (500 GB) with OS 10.6 and 10.7 partitions, new memory (2 X 2GB strips), new PRAM battery, new optical drive also to extend the life of the mini for my granddaughter who wants to use it with her iPad mini.
Lion OS (10.7)
If you order Lion from the Apple Store you will get two emails after about 24 hours, one with a password, the other with a content code. If you try and use the content code to download to your upgraded mac mini it will tell you that your computer does not support Lion and it will not download the software. If you try it on another machine it will say that the code has been redeemed. You will have to call Apple and ask for another code then wait another 24 hours.
Download onto an external drive attached to a compatible Mac. I'm using my Mac Pro. Before you can use Lion on your Mac Mini you will need to erase a file that lists your Mini as being incompatible. The filename is platformsupport.plist. Delete this file and clone the installed OS (on the external drive) to your Mac Mini. Run the bios updater next if you want to use 3GB RAM. A few hoops to go through to get to the finish line! Good luck!