Hurin's debt?

[Hot, crispy nice hobbit]

I was rereading the Silmarillion for the second time this week, and I still could not figure out why Hurin chose Nauglamír of all jewels to repay Thingol:

'Receive thou thy fee,' he cried, 'for thy fair keeping of my children and my wife! For this is the Nauglamír, whose name is known to many among Elves and Men; and I bring it to thee out of the darkness of Nargothrond, where Finrod thy kinsman left it behind him when he set forth with Beren son of Barahir to fulfil the errand of Thingol of Doriath!'

Was there really any significance behind that jewel? If Hurin is of darker nature, he might have dug up the Dragon-helm from the wastes of the Woodmen's dwellings: the Dragon-helm, at least, is directly related to the plight of his family. Why dig up a jewel that did not enter his story at all?

It puzzled me a great deal why Hurin chose the Nauglamír if he scorned Thingol and Melian so much at that time, and wanted to bring them something to remind them of their deeds.