What do you mean by bird synonyms? Do you mean alternative English names? Current alternative scientific names that are used in different taxonomies? Or historical scientific names, such as names given by authors in the nineteenth century?
Like Desmond, I'm not sure if you are looking for alternative English names or scientific ones, and if the latter, just how comprehensive a synonymy you are after. Most of the resources already given by others are lists rather than synonymies, though a few do give some useful information, especially foreign language equivalents to English names. If you are looking for scientific synonymies, I’m not aware of anything approaching a complete synonymy online. To do the job properly you’ll need some printed works as well – many now available online. A good starting point to the older synonyms is the Catalogue of Birds in the British Museum (1874-98, 27 vols; available at http://www.biodiversitylibrary.org/bibliography/8233#/summary). For palearctic birds you then should trawl through Hartert (http://www.biodiversitylibrary.org/bibliography/14175#/summary) and Vaurie’s 2 vols of Birds of the Palearctic Fauna, with other stuff as needed. Peters’ Checklist of the Birds of the World (1931-1987, 16 vols; http://www.biodiversitylibrary.org/bibliography/14581#/summary) is indispensable, and probably your best bet. For a one-volume compilation of the more recent alternative English and scientific names in use worldwide you could look at MG Wells’ 1998 World Bird Species Checklist.