Well...That depends. If you mean only objects with names like 'Sirius', 'Archernar', 'Lundmark's Nebula','Stefans Quintet', then the number of objects with names like these is around 1000 or less. Most of these are the names of the brighter few hundred stars. But, if you are not a purist and are willing to regard some catalog names as proper names, then the answer is in the 10's of thousands.
Examples of such objects with 'quasi' proper names based on a catalog are:
Wolf 359 Lalande 21185 Krugar 60 Eta Carina H and Chi Persi Zeta Ophiuchi Gleise 290 Sharpless 109 W3 40 Eridani B NML Cygni Centaurus A Omega Centauri Cygnus X 30 Doradus