Verbal Reasoning!

[Legate of Amon Lanc]

Well, if I recall correctly, the Dwarven mines were in the north of the mountains, I'd have to check it but I believe there is not explicitely said where, but it looks like it was somewhere on the edge of the mountains of Angmar or something like that... so actually, also part of the Misty Mountains.

[The Might]

No...the mines were still in the Blue Mountains, and afterwards he lived on the shore of the Icebay of Forochel.

[Beregond]

The Might, you're right, but for the question note that I said Hithaeglir instead of Misty Mountains...

[Legate of Amon Lanc]

Okay, I looked it up, and it says:

But King Arvedui held out upon the North Downs until the last, and then fled north with some of his guard; and they escaped by the swiftness of their horses.
For a while Arvedui hid in the tunnels of the old dwarf-mines near the far end of the Mountains, but he was driven at last by hunger to seek the help of the Lossoth, the Snowmen of Forochel.

Since the North Downs are about half way from the Misty to the Blue mountains (and maybe a little bit closer to the Misty), and there are not any Mountains mentioned earlier (just "Mountains"), we can only speculate what Mountains these were. But I was always under the impression that it was the end of the Misty Mountains, or the Mountains of Angmar, because to get to Ered Luin, Arvedui would have to cross Lune and before, go around - or through - Emyn Uial, which doesn't seem logical because he could as well flee to Mithlond and not complicate the matters any more. But we are also told about "swiftness of horses", and going through or around Emyn Uial on horses is not much of a bonus, however on the vast plains between North Downs and the Misty Mountains (or mountains of Angmar) the horses could use their speed well, as later were by the Witch-King himself in his pursuit by Glorfindel.
Also, there is some strange sort of mountains or hills depicted on the LotR map just west of the mountains of Angmar, and since it seems unlikely that the King will on his flight willingly get near the enemy's capitol of Carn Dum, the Dwarven mines could as well be there.

EDIT: Cross-posted. Hmm...

[Beregond]

I would not have thought he would get even that close... and I assumed it was the Mountains of Ered Luin, but now you've left me wondering if that's true. Ered Luin was what I was looking for anyway, the Sindarin name of the mountains that I thought Arvedui hid under, but since I may be wrong, it was a poor question!

Legate, I will say you got it because you first mentioned the old dwarf mines.

[Legate of Amon Lanc]

Well, this is sort of troubling, since if you were looking for Ered Luin, then I'd say Miggy got it. But... probably if we don't know for sure where the mines were, and if you don't want to use your right as the riddle-poster to say "for MY riddle, THIS answer is correct", then... ah, whatever, let the thread move - it will surely swap between us more times, so...

Isildur is to Gil-Galad

as

Túrin is to _______

[Beregond]

Ah, yes, sorry if I was a little unfair, Miggy, wasn't intended.

I would say Turgon, because he was the reigning High King of the Noldor during Turin's lifetime, just as Gil-Galad was the High King of the Noldor for the most part of Isildur's lifetime.

[Legate of Amon Lanc]

Beregond: Fine, but it's not that general (many people lived at the same time - this works both ways, it is more personal).
Mig: Good idea, you did well to bring Sauron between them, though it's not what I was looking for.

It's personal - so it relates to the people who really were in contact.

[The Might]

maybe Finwe
Isildur killed Sauron and so avenged Gil-galad
It has been prophesized that Turin would kill Melkor, and would thus avenge Finwe
Why I picked him is because both he and Gil-galad were High Kings of the Noldor