The only alternative that comes to my mind is Zotero:

• It is open-source,
• it comes as a standalone application or as a web-based version with Firefox, Chrome and Safari connectors,
• it integrates with Word or OpenOffice,
• it syncs with the Zotero server,
• it has BibTeX export,
• and more.

The Zotero standalone client is cross-platform and open-source (AGPL licence), and it can be run on its own or synchronized with the web version. The web service is free to use up to a fixed storage quota, with paid storage available, and there is an open-source implementation of the dataserver available if you want to roll your own. The local client stores its data in SQLite format so in principle your data is not locked in, but the database is relatively hard to trawl externally; however, since the client is open source there are relatively few future-proofing concerns.

Docear

Haven't tried it yet but https://www.docear.org/ seems to stand out as one of the leading open-source alternative to mendeley.

Docear is basically a marriage of JabRef and Freeplane. It uses JabRef as a backend for its reference management and Freeplane to organize references, annotations you make in the pdf, and any other information (including images, links and crossreferences) in a mind-map.

As the original author of this answer pointed out it is indeed open-source—licensed under the GNU General Public License.

I really liked Mendeley's potential but got frustrated with both their pricing model (maybe I just never learned how to use the software correctly) and it consistently butchering imported BibTeX entries.

I've been a pretty happy BibDesk user for a long time, it is true open source software, but unfortunately it has not been ported outside of the OS X environment, so this is only a qualified answer.