[Legolas]

Manwe, it is what we properly call Rhûn.

Aragorn to Boromir:

I have crossed many mountains and many rivers, and trodden many plains, even into the far countries of Rhûn and Harad where the stars are strange.

Denethor to Pippin:

And in my turn I bore it, and so did each eldest son of our house, far back into the vanished years before the failing of the kings, since Vorondil father of Mardil hunted the wild kine of Araw in the far fields of Rhûn.

In 'The Field of Cormallen' chapter:

But the Men of Rhûn and of Harad, Easterling and Southron, saw the ruin of their war and the great majesty and glory of the Captains of the West.

From Appendix A, I - The Númenorean Kings, iv - Gondor and the Heirs of Anárion

In his day Gondor reached the summit of its power. The realm then extended north to Celebrant and the southern eaves of Mirkwood; west to the Greyflood; east to the inland Sea of Rhûn; south to the River Harnen, and thence along the coast to the peninsula and haven of Umbar.

From Appendix A, II - The House of Eorl

And wherever King Elessar went with war King Éomer went with him; and beyond the Sea of Rhûn and on the far fields of the South the thunder of the cavalry of the Mark was heard, and the White Horse upon Green flew in many winds until Éomer grew old.

It literally means 'east,' but yes, it is the proper name. Names do usually mean things about their bearers' nature. Mithrandir means ' Grey Wanderer,' but it is still his name. Ulmo means 'He Who Pours' but it is still his name. Not until (relatively) recently in the western world has it become popular to give names without giving consideration to the meaning behind them.