shortDOI is a URL shortening service that takes DOIs and converts them to short URLs such as http://doi.org/bb6, which is nice for emails and Twitter. You can add the bookmarklet by dragging this link to your bookmarks: shortDOI. It will try to find the DOI in the current page and direct you to shortDOI.
A shortDOI URL is probably more persistent than, say, bit.ly, as it's backed by the organization that maintains the DOI infrastructure. However, if doi.org would go down, you could always use a search engine if you have the original DOI, but the shortDOI URL will be worthless.
Update 24.02.2011: Use some majority voting to find the right DOI.
Update 01.03.2011: Expand the list of allowed characters. Does anybody know which characters can be part of the DOI?
Update 29.07.2015: Be strict about having a prefix and suffix. First check for the "citation_doi" meta tag before looking in the rest of the document.