(Sorry for forgetting to select a category)
There are a few details I would like to specify for this feature:

h2. Sub-features of Achievement System

* Admin Interface - For Admins to manage Achievements
* Display Area for Achievements on User Page - Viewable by Other Users
* Display Area for Achievements on user's My Page (customizable via "Personalize My Page") - Viewable by Self
* "All Achievement" page - Viewable by All
* Achievement Rankings - Encourages Achievement Competition

h2. Elements of an Achievement

An achievement may contain these elements:

For Users
# Name|String: A single-line text string that defines the name of the Achievement, e.g. "First Blood";
# Description|Text: A multi-line text string that describes the Achievement, e.g. "Completed your first assignment";
# Icon|Image: A simple image that flavors this achievement;

(All the elements above are defined/uploaded by Admins)

For Admins
# Requirement|Text: A multi-line text string that describes the triggering condition or mechanism that how this achievement should be rewarded to a user. This element is for Admins, so it might be a less user-friendly description. e.g. "User's Assignment Status changed from Assigned to Completed";

h2. Admin Interface

To manage the Achievements, there should an interface or control panel for the Admins. This interface should provide functions such as:
# Add or Delete Achievements;
# Edit Achievements, defining or changing its elements stated above;
# Manually reward/remove achievements to/from users (individually or batch);

h2. Working Mechanism of this "Achievement System"

Here goes a simple Workflow demostration:
# Admin Adds/Defines Achievements
# Display in "All Achievements" Page
# Users meet the requirement of certain Achievements
# Admin reward the corresponding Achievements to Users
# Completed Achievements display in User Page and user's My Page
# All users with Achievements are ranked in a Ranking page

h2. Other Notes

# Since it is difficult to customize how the program could recognize when a user meets the requirement of an Achievement, so all Achievement can be manually rewarded to users by Admins. (Which means, no automatic Achievement triggering)
# Since there is no "automatic triggering", it is better that there is a "Apply for Achievement" feature for the users, so they can ask for the reward when they finished their tasks; and a "Achievement Reminder" for the Admins, so they can efficiently reward achievements to those who deserve;
# Better there's an "Achievement Center" where all Achievement-related actions are gathered. (however the displays should also be on user page and their "my page")
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I don't think it's a general feature that will be interesting for everybody using Redmine, but it would be great in some cases. So it's a perfect fit for a plugin! Actually your description is pretty complete, thanks for that, it should be a good starting point.
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