The Silmarillion only gives this :
And it is told that in that time Daeron the minstrel of Thingol strayed from the land, and was seen no more. He it was that made music for the dance and song of Lúthien, before Beren came to Doriath; and he had loved her, and set all his thought of her in his music. He became the greatest of all the minstrels of the Elves east of the Sea, named even before Maglor son of Fëanor. But seeking for Lúthien in despair he wandered upon strange paths, and passing over the mountains he came into the East of Middle-earth, where for many ages he made lament beside dark waters for Lúthien, daughter of Thingol, most beautiful of all living things.
One is left behind wondering about the poor minstrel’s anguish…
It is certain though, that he endeavoured to follow Lúthien after she had laid herself down in the grass of Doriath and passed away to Mandos, and that he besought a way to follow her and try for her release; but the Enchanted Isles repulsed him and made passage impossible.
How many days must he have lingered about the shipyards of Brithombar and Eglarest, without food or sleep, bitterly accusing the Doom of Mandos of cruelty? He sang his complaints to the rocks and mountains, melting the hearts of wolves and bears, moving the birch trees of Nimbrethil from their stations.
He held himself aloof from womankind, dwelling constantly on the recollection of his sad mischance. But then one day, he came upon the maids of Melian, whose advances he had always repulsed during his time in Menegroth. They bore with him as long as they could; but finding him insensible finally one of them exclaimed : "See yonder our despiser and the betrayer of Tinúviel!", and threw at him a sharp rock.
Though the projectile, as soon as it came within the sound of Daeron’s harp, fell harmless at his feet. So did also the stones that the other maids consequently threw at him. But the livid she-elves raised a scream and drowned the voice of Daeron’s music, and then the missiles reached him and soon were stained with blood. The frenzied maids tore him limb from limb, and threw his head and his harp into the river Nenning, down which they floated, murmuring sad music, to which the shores responded a plaintive symphony.
It is said that Uinen herself gathered up the fragments of the minstrel’s frail body, once it reached the open sea, and buried them at Tol Balar, where the nightingale is said to have sung over his grave more sweetly than Tinúviel had ever sung in Neldoreth. His harp was placed by Varda amongst the stars. His spirit passed to the Halls of Mandos, and it is said that there he forever sought out the spirit of his beloved Lúthien, but never found her as her gentle spirit had left the confines of the world until Arda itself would be broken and remade.
There in the Halls of Mandos he remained, alone, until the end of Time, after which (hopefully) he would be reunited and reconciled with the most fair and most beloved of all Children of Illúvatar that ever lived.