This question feels off topic to me. There is a lot of information in the question that isn't relevant, including:
- This is for a piece of fiction
- This is for a piece of hardware that is 10 years old
- The author's limited knowledge of hardware from 10 years ago
- "Function" being proposed by this fictional hardware
The heart of the question boils down to:
- "[W]hat was good back then?"
- "[H]ow would [Linux] run on a '05 Gaming tower?"
The author provides some stats to work with, including RAM, storage space and possible CPU options.
With all of the above, I feel this question is either too broad or unclear. It's too broad because both of the above questions have a lot of answers. The first one is impossible to answer in the context of this question too, because the context is fictional. How are we to know what "good" means, in terms of running a time machine. My time machine is either a Delorean, for the cheap time traveler, or a TARDIS for the clever ones.
As for the "how would Linux run" portion, this can't really be answered without knowing what hardware is used. The same holds true today. "Will Linux run on my machine?" Sure...but I don't know how well until you tell me what the machine's specs are.
The question is unclear because of the context. The purpose of the question is to find hardware that can effectively run a fictional machine. Since it's fiction, I'd think a small amount of hand waving is allowed when describing the computer.
For these reasons, I voted to close it as unclear.