This is a fantastic idea! I do not think it needs to be the time stamp saved on the server. Rather, just use the client time. It's only purpose is to show the log progression to the player. Who said what and when, relative to his/her own time frame (PC or PS3 time).

There is a chance that the user may change their regional setting or actual clock during play, but this is a very small chance I think. The great majority of users will not be doing this. Just add a note to the release that mention the clock display in the log window is always the user's system time.

This should have little impact on the client and server load. Allowing the log to save will be a bonus!