‘Old Norse’ is a term very broad in both time and space. In Viking times, it is clear enough that people from all over Scandinavia, and from the peripheries of the Viking world, i.e. Norwegian settlements in the North Atlantic, Danish settlements in Britain, and Swedish settlements in the Baltic and south to the Ukraine, were mutually intelligible to each other in their speech.
But this is a huge area; combined with the general lack of central power (and thence the lack of standardized speech), dialectal variation of this common speech must logically have been enormous. In counterweight, the Nordic people of that time did seem to travel widely, with their renowned sailing skills, which would serve to standardize speech. Post-thirteenth-century, when Old Norse probably ceased to be a practical term in Scandinavia, dialectal variation must have increased with the rapid decline of Nordic navigation skills and end of the mobility of the Viking era.
But as I said, ‘Old Norse’ is a broad term not only in space, but also in time. Not including Old Icelandic into the equation, Old Norse would seem to have a lifetime of roughly half a millennium. If we multiply that with the aforeknowledged variation in space, we should conclude that any thought of "standard" Old Norse is hopeless.
But this is generally, if not universally, accepted by academics. The ‘reconstructed pronunciation’ used today is of mere practical value, based on a mediation of the "most standard" sounds, and to some degree on the work of contemporary phoneticians, "native speakers" (though of course just of their own dialect, in that time and place). The perhaps idealized vowel length distinction, with entirely no distinction in quality between graphically equivalent long and short vowels, is the hallmark of the reconstructed pronunciation.
Whatever speech standard we devise for Old Norse, the most important criteria is that it differentiate all the different graphs used in the ‘standardized spelling’ (orthography was another part of the language that varied enormously from time and place, especially in accent marking).
In any case, I believe anyone knowledgeable of Old Norse and good phonetics should feel free to theorize his/her own pronunciation standard according to his/her best academic capacity. That is what I should like to entertain you with, i.e. an alternate pronunciation standard falling closer to my own convictions in Nordic philology. I should comment that since my partner in the "Old Norse for Beginners" course, Haukur Þorgeirsson, expresses positive consent with this pronunciation scheme, I present it specifically in connection to our course, as a suggested alternative to our students.
The following table presents the ‘standardized spelling’ graphs to be presented, along with SAMPA transcriptions of the standard pronunciation, then transcriptions of my alternative pronunciation, and finally rough English (or other) approximations, for clarity:
Character Standard Alternate Approx.
á [A:] [O:] aw (law)
a [A] [A] Brit –ar (star)
é [e:] [e:] e (pet), long
e [e] [e] e (pet)
í [i:] [i:] ee (fee)
i [i] [I] i (pit)
ó [o:] [o:] Germ oh (wohnen)
O [o] [o] Sp o (hijo)
Ú [u:] [u:] ue (Sue)
U [u] [U] oo (foot)
Ý [y:] [y:] Fr u (rue)
Y [y] [Y] Ice u (kuti)
ö (o-tail) [O] [O] aw (law), short
œ [2:] [2:] Fr eu (feu), long
ø   Fr eu (feu)
æ [E:] [E:] ai (fair)
ei [Ei] [Ei] ay (day)
ey [9y] [9y] Fr euil (feuille)
au [Au] [Au] ow (brow)
Consonants = as in ‘standard pronunciation’.
The differences in the system are that
a) the high vowels (í, i, ú, u, ý, y) are differentiated both by quantity and quality, the shorter ones being slightly laxer (less close) then the long ones. This is the general trend in Germanic vowel systems, and I dare say the general trend in the world’s languages overall.
b) meanwhile, the mid vowels are differentiated only in quantity; they cannot be differentiated further without conflicting with other vowels.
c) ‘a’ and ‘á’ are differentiated also in quality, ‘á’ being a more rounded back sound. This is a known-to-have-happened sound shift in Old Norse, cf. the Scandinavian reflex ‘å’, pronounced as [O] (length usually depending on environment). In Faroese and Icelandic the sound further diphthongized, becoming [Au] (perhaps initially [AO]) in Icelandic and [OA] in Faroese. It is really just a matter of timing, whether one pronounces ‘á’ in this way, since the [O:] pronunciation is perfectly natural of later ON. I choose it because it also has the advantage of being easier/more natural to many Germanic speakers, especially English and Scandinavians.
Overall, the differences between this scheme and the standard one are by no means radical. I mean only with this effort to suggest a less artificial method of pronouncing ON. The artificial feel to the reconstructed pronunciation makes it rather difficult to pronounce, whether for native Icelanders, Faroese or others.