ok kòlluðu þeir á Þór, ok jafnskjótt kom hann, ok því næst fór á lopt hamarrinn Mjòllnir.

and they called to Thor, and like a shot he came, and next thing the hammer Mjöllnir rose aloft.

-- Snorri Sturlusson, Gylfaginning, ch. 42

Philology of Mjöllnir and Mjölnir

This name is usually known in the form Mjöllnir, but our resident Icelander paladeen uses the form Mjölnir, and assures me it's the one current in Iceland today. What is the difference? Is one right and the other wrong?

No, both are correct. The two-L form is more common in the ancient texts of Old Norse, from where our tales of Thor primarily come. The Old Norse of the Icelandic sagas of 600-700 years ago is almost identical (in writing, at least) to modern Icelandic, and Icelanders can read them more easily than English-speakers read Shakespeare. But there are very slight differences in spelling. This appears to be one of them.

Here's how Professor E.V. Gordon in his Grammar of Old Norse explains it, in a note on p. 197:

Mjòllnir: the name probably means 'the shining one'; cf. the cognate Russian molnija 'lightning'. It has also been interpreted as 'crusher' (cf. ON mylja 'crush'), but the name is usually spelt in MSS. with double ll, which is then difficult to account for.
Let's pick that apart. First, I have had to use ò instead of the correct character, which I can't depict in HTML. The actual letter was an O with an ogonek, a hook underneath it open to the right. (lj tells me its ǫ so if you can see that the name is Mjǫllnir.) This was pronounced as in yawn, and normally arose from a when there was a u or w in the following syllable. By about 1300 it had shifted pronunciation to more like the vowel in yearn, and in modern Icelandic this sound is written ö, as in German and Swedish.

But in the group it normally arose from an earlier e before u or w. The exact sequence of changes is unknown, but compare mjòðr 'mead' and mjòl 'meal (flour)', where English preserves the original e. So what this suggests is that an earlier form of the name might have been *Melluni-. (The final -r is a case ending, like Latin -us.) And where did that come from?

The Proto-Indo-European stem *MEL-/*MOL- meant 'soft', and is widespread in European languages: English words derived from it include melt, mild, mollify, mollusc. Extended to the meaning 'crumble' it occurs in mole (the animal) and mould (in the sense of earth). Extended to the causative meaning 'grind' it occurs in mill and meal (flour) and mallet and molar (tooth). It is this latter extension that is appropriate for the name of a hammer: related to the Norse verb mylja 'crush, mill'.

English mill comes via Middle English miln, myln from Late Latin mulina, Latin molina. Etymologically it has only one L. This is important: if the Norse name came from this 'grind' root it should also have only one L. Words don't simply grow double letters for no reason: how would an original Mjòlnir change to Mjòllnir? The circumstances were few and limited where doubling occurred in Old Norse, and this wasn't one of them. This is what Gordon means by saying this derivation is difficult to account for.

It is, however, among the circumstances where shortening of long consonants could occur. This was more widespread: *fagrr became fagr 'fair', *jarll became jarl 'man, earl', and vettr became vetr 'winter'. So it is likely that the original was Mjòllnir and this has become Mjölnir by normal phonetic processes.

I can't find anything in other European branches related to Russian молния 'lightning' and мелькать 'to gleam', so we can't really be confident about connecting it to a distant cousin like Old Norse, but it's what's left after we rule out the more common *MEL-/*MOL. Russian has words from that root too: words for mill and hammer, mallet. Today Russian has no intrinsic double consonants, but an earlier root with double L is not ruled out.

The simplest thing we can say is that we don't know what Mjòllnir means, but here is some of the evidence for and against some of the possibilities, as I understand them.