Please provide the single best option you are aware of.
HTML Tidy does a very good job on Word 2000 HTML, but I'm not sure how well it works on newer Word output.
Dreamweaver has a "clean up word HTML" option. Granted, I know it isn't perfect, but it also the only thing I have worked with.
Rather than cleaning up Word's HTML you could generate HTML directly from the Word document using Abiword. (wv is now deprecated in favour of Abiword; it's basically been absorbed.)
An example:
AbiWord --to=html file.doc --exp-props="html4: yes"
See more in the documentation.
Word 2007 has a "publish > blog" menu item on the Office menu (top left corner).
Using this feature seems to do an incredibly good job of cleaning the HTML, far better than any of the other HTML exporters built into Word (like "save as HTML Filtered").
I have actually set up a bogus free blog somewhere just to use this HTML-cleaning capability. Most long articles on Joel on Software originated in Word 2007 and was published to a fake blog just to clean up the HTML.
Edit: as pointed in comments, be sure that you enter a title for the fake post. If you don't, Word will show a generic error "Can't publish your post"
Not exclusive to Word documents, but it is free (and open source). You might try Tidy: http://www.w3.org/People/Raggett/tidy/
There is a tool I wrote awhile back, it's a web application for converting Word DOC files to HTML. You just upload the .doc file and you get this interactive view of the conversion with a bunch of different options to tweak it. It's up here if you want to give it a try:
I found this tool: http://opilsoft.com/doctohtml.html, Opilion Software DocToHtml, it does a very good job. It produces the smallest HTML among all utilities that I tried.
Some online solutions:
A free tool that removes the excessive tags and clutter from Microsoft Word generated HTML documents, leaving basic formatting intact. File sizes are greatly reduced, and the returned HTML is easier to read, revise and employ.
This site cleans up the HTML for any pasted document. It applies a number of filters to fix various things that Microsoft Office puts in its HTML, and gives you a nicely formatted result that you can paste directly into a web page or content editing system.
A tool to strip Microsoft’s proprietary tags and other superfluous noise from Word-generated HTML documents, leaving all the basic goodness intact. The service is free of charge for documents up to 20Kb in size. For larger files, subscriptions are available, proceeds from which go to keep your host’s dogs in biscuits and squeaky toys.
There is a solution for this on Coding Horror: Cleaning Word's Nasty HTML.