There is a [addr:housename][1] tag which you could use to enter house names.
names. The [name][2] tag could also be used, but I guess it is more appropriate to use it on POIs.
Rendering is another issue. [Mapnik][2], [Mapnik][3], OSM's main renderer, only renders house numbers / names if they roughly fit inside the rendered house outline. outline (and don't conflict with surrounding captions). If the name is too long, it won't show up on a map rendered by mapnik. [Osmarender][3] [Osmarender][4] instead is a bit more tolerant. You can activate it by selecting the corresponding layer via the blue *+* in the top right map corner or by adding *&layers=O* to the URL. Maybe the name of your buildings will show up there. Besides, both mapnik and osmarender will need some time to render your updated map, especially at the weekend.
[1]: http://wiki.openstreetmap.org/wiki/Key:addr:housename
[2]: http://wiki.openstreetmap.org/wiki/Name
[3]: http://wiki.openstreetmap.org/wiki/Mapnik
[3]: [4]: http://wiki.openstreetmap.org/wiki/Osmarender
