There are bound to be others who will disagree with me but I will give you MHO. I work at a university in desktop and network support so I do have some knowledge on the subject. A 500mhz system would be one that we would recommend replacing. In your post you mentioned that the seller seemed to be avoiding telling you about the memory. He may not even know. This is another reason I would stay away from this system. A computer's speed and ability to run programs does not depend solely on the processing speed but the RAM (or memory) as well. You were right to ask about this. If it had 64MB (or IMO - anything less than 256mb) of RAM it could be frightfully slow. You also mentioned it would be used for emailing which requires a connection to the internet. So also consider what operating system is installed on this unit. If it is Windows 2000 or XP it is vital that you are able to download and install the windows updates necessary to protect your machine from worms and other attacks. If you are planning to use a dial-up connection this can be painful enough, but coupling that with a system that is outdated with a slow processor and insufficient RAM and you might just lose your mind! And if a system like this becomes infected with parasites (spyware) it will be excrutiatingly slow.
Having said all of this, I do have a 500mhz system in my home that I wiped and loaded kids software on for my daughter to use. It suits her needs just fine (she is only 5). I DO NOT have this computer connected to the internet however and I also have installed 256mb of RAM in the computer.
So - I would say it would be vital to know how much RAM is installed in the system, what operating system is installed and was it wiped with a clean install of the OS or was it used for years and then put up for sale. If it was a clean install and the system had 256mb of RAM and you (or your mom) are willing to take the time necessary to keep the OS updated by downloading and applying the updates and are just looking for unit to play games (very simple, non-graphics intensive) and e-mail for maybe a year, then it would be ok. Otherwise steer clear.
I hope this helps you. No flames please - I am giving the poster my opinion based on my experience.