Dreamweaver does that by maintaining link states internally in a proprietary template system so link checking can be auto-corrected if link dependencies fail. Eclipse is noteworthy because it's Java based and runs on any platform making it easily scalable across nearly all platforms. Not because of its plugins.
Now don't take this the wrong way, I'm being brutally honest and factual here and it not an attack on YOU personally:
You might have a hard time finding plugins that auto-magically fix link dependencies because running a standard link checking report is usually all one needs to find/fix broken links. You described your site as "whole lot of files in different folders" which meant to me you have a ton of repetitious, broken links that need addressing after making changes. It also proves, without a doubt, that the site was poorly organized from the outset and it fails a lot of SEO best practices and standards in templating. For example, the menu system is not a centralized script and is in many folders, or articles with links to other articles are duplicated, or links in content area also in sidebars, headers or footers but not included dynamically. Plus many other reasons.
So in cases of badly organized sites with poorly defined or lack of coherent templating, the real fix is to run a detailed link checker report and find broken links and fix them, using global search and replace. Or, taking the time to redesign the site and improve SEO indexing and performance, if the client is willing to spend the money. If not, point it out to them. If you can't then the moral of this story is you can't easily automate a fix for a broken web site.
I want to be crystal clear this is not blaming or patronizing you in any way or saying you caused this. I thought carefully before responding in this manner, and after checkingt a list of Eclipse plugins at
http://en.wikipedia.org/wiki/List_of_Eclipse-based_software and a few minutes Google searching. I failed finding anything, then realized you might be looking for something that doesn't exist to satisfy unfair client demands on something likely broken before you even saw the software. So with those assumption in mind, consider a quality link checker report as utility #1 in your toolbox in this situation.
Wouldn't hurt to check back here in a few days if time permits to see if any other suggestions from others, as I admit my assumptions are many. But I have this feeling I am right on track with this.... been there, done that.
Cheers, and thanks for listening.