Best module for administering comments?

The first step is to install a good CAPTCHA module. A CAPTCHA on the comment form will weed out most of the offending comments. The Sun also requires that comments be approved before they go on the website.

Other than that, we wrote a simple module that adds a menu item to the top of the Admin menu. The menu item displays the number of unapproved comments and links to the comment approval queue. (I got some code checked in to Drupal 7 so that menu item for the comment approval queue displays the number of unapproved comments; you just need to move that menu item higher up in the menu to get the same effect.) That menu item helps us keep on top of unapproved comments as they come in.

There's also been talk of giving some editors email notifications when a new comment gets posted, but noone's found time to do that yet.

In terms of tracking a user's comments, you could always play around with the Views module and set up a comment view that displays all the comments for a given user.

I'd still recommend CAPTCHA if you haven't installed that. I feel like if you filter out the spammers, the number of problematic users tends to be low enough that you can remember who they are. But if you really need to track problematic users, try setting up the comment-based view.

But in any case, there's an interesting assumption you've made, and it's that only Web staff should be responsible for moderating comments. I don't know how much power you have to change the status quo, but it's an assumption I don't entirely agree with...

I'd look at what you can get using views to display comments, and vbo to then moderate comments (and, if needed, users) --

The flag module integrates well with views; just be sure to set up the relationships for the flag module via the views UI.

I would say it heavily depends on your site, your policies and your readers.
On http://www.zeit.de (a major german newspaper) we handle comments this way:

All new Comments go online immediately. At the same time they are
put in moderation queue, which is read by our moderators.
Each comment is viewd by a moderator using the moderation module [1]
we developed to speed up the process.
If the moderators find an offensive Comment, the text is removed
from the comment, replacing it with a set of standard notices like
"Please don't be offensive".
No comments are deleted at all in order to give the user the transparency
what happens to their content and why.

In addition to this, we developed the memo module [2] where deleted
parts of the comments are put by the moderators. This is necesessary if
arguments with the user come up, wether the moderation is justified or not.

As a third tool against trolls, we developed the ipnotification module, which
sends a message to our moderators, if a user with a given IP-Part, User-Name
Part or Email-Part tries to login or post a comment.

As only registered Users are allowed to comment we have no problem with
automatic-spam. So we do not need Captchas or Mollom yet.

Of course our solution needs a feed moderators, but I think, if you seriously run
a newspaper-site an you want user-interaction you really should make sure,
that some really reads what your readers are writing. :)

I hope I could help you somehow. If you have any questions, I'm here.

[1] http://drupal.org/project/moderation
[2] http://drupal.org/project/memo
[3] http://drupal.org/project/ipnotification

Ah. Yeah. We also do flagging.

Currently we use the regular Flag-Module.
Every User (even unregistered) can flag a comment as "unapproriate".
We created a view that lists all Comments with more the five flags.
This works quite fine, but still we are going to change that, to work like this:
If you want to flag a comment, you have to leave a notice, why you
think this comment is unappropirate.
We'll remove the 5 flags barrier than.

I'll keep you up to date, wether this works better or not. :)