[Maédhros]

Originally Posted by William Cloud Hickli

But there was no way to have Hurin appear at Menegroth without 'editorial invention:' Tolkien pere never wrote it.

That is not right. From the Book of Lost Tales II:Turambar and the Foalókë

Now Húrin caused his followers to bear this gold to the halls of Thingol, and they murmured at that, but he said: "Are ye become as the drakes of Morgoth, that would lie and wallow in gold and seek no other joy? A sweeter life shall ye have in the court of that king of greed, an ye bear such treasury to him, than all the gold of Valinor can get you in the empty woods."
Now his heart was bitter against Thingol, and he desired to have a vengeance on him, as may be seen. So great was that hoard that great though Húrin's company might be scarce could they bear it to the caves of Thingol the king, and some 'tis said was left behind and some was lost upon the way, and evil has followed its finders for ever.
Yet in the end that laden host came to the bridge before the doors, and being asked by the guards Húrin said: "Say to the king that Húrin the Steadfast is come bearing gifts," and this was done. Then Húrin let bear all that magnificence before the king, but it was hidden in sacks or shut in boxes of rough wood; and Thingol greeted Húrin with joy and with amaze and bid him thrice welcome, and he and all his court arose in honour of that lord of Men; but Húrin's heart was blind by reason of his tormented years and of the lies of Morgoth, and he said: “Nay, O King, I do not desire to hear such words -- but say only, where is Morwen my wife, and knowest thou what death did Nienor my daughter die?” And Thingol said that he knew not.
Then did Húrin fiercely tell that tale, and the king and all his folk about him hid their faces for great ruth, but Húrin said: "Nay, had you such a heart as have the least of Men, never would they have been lost; but lo, I bring you now a payment in full for the troubles of your puny band that went against Glaurung the drake, and deserting gave up my dear ones to his power. Gaze, O Thingol, sweetly on my gifts, for methinks the lustre of gold is all your heart contains."
Then did men cast down that treasury at the king's feet, uncovering it so that all that court were dazzled and amazed – but Húrin´s men understood now what was forward and were little pleased. "Behold the hoard of Glaurung," said Húrin, "bought by the death of Nienóri with the blood of Túrin slayer of the worm. Take it, O craven king, and be glad that some Men be brave to win thee riches."
Then were Húrin's words more than Thingol could endure, and he said: "What meanest thou, child of Men, and wherefore upbraidest thou me? Long did I foster thy son and forgave him the evil of his deeds, and afterward thy wife I succoured, giving way against my counsel to her wild desires. Morgoth it is that hates thee and not I. Yet what is it to me -- and wherefore dost thou of the uncouth race of Men endure to upbraid a king of the Eldalië? Lo! in Cuiviénen my life began years uncounted before the first of Men awoke. Get thee gone, O Húrin, for Morgoth hath bewitched thee, and take thy riches with thee" -- but he forebore to slay or to bind Húrin in spells, remembering his ancient valiance in the Eldar's cause.
Then Húrin departed, but would not touch the gold, and stricken in years he reached Hithlum and died among Men, but his words living after him bred estrangement between Elves and Men.

Changes for the better understanding of the text. The original names have been changed to the latter ones. They are in bold, in the text.

Úrin > Húrin, Tinwelint > Thingol, Melko > Morgoth, Mavwin > Morwen, Nienóri > Nienor, Glorund > Glaurung, Hisilómë > Hithlum, Palisor > Cuiviénen.

However, if one wants to be literal.
From the Book of Lost Tales II: §2. Places and peoples in the Tale of Tinúviel

The picture of Menegroth beside Esgalduin, accessible only by the bridge (The Silmarillion pp. 92 -- 3) goes back to the beginning, though neither cave nor river are named in the tale. But (as will be seen more emphatically in later tales in this book) Tinwelint, the wood-fairy in his cavern, had a long elevation before him, to become ultimately Thingol of the Thousand-Caves ('the fairest dwelling of any king that has ever been east of the Sea'). In the beginning, Tinwelint's dwelling was not a subterranean city full of marvels, silver fountains falling into basins of marble and pillars carved like trees, but a rugged cave; and if in the typescript version the cave comes to be 'vaulted immeasureable', it is still illuminated only by the dim and flickering light of torches (pp. 43, 46).

One could say that at the time of the meeting of Húrin and Thingol in Menegroth, the name Menegroth was not in existence.