Labranda refers to labrys (=double bladed axe). It has an ancient history, but was conquered by the Persians (after the nearby Milas) and had to be rebuilt. This was done by the satrap Mausolos (377-353 v. Chr.) and his two brothers after him. He saw it as his ancestral sacred shrine. In the Wikipedia it is suggested there existed a golden double axe in Priene that somehow got in the hands of the Labranda people, who kept it in the Temple of Zeus Labraundos.
The setting is beautiful, high up in the hills, where you arrive after a trip along a road that is in need of repair. You have great views over those hills, and if you are in luck the very friendly guard (who seems to live on the site) will show you around. To be the only visitor (as I was in March 2007) is an experience not easily forgotten. The more usual visit seems to be: arrive by bus, be shown around in 20 minutes, and off you go.
I show roughly my walk over the site: arrive, gradually climb till the grave higher up, and then down again.