I have to go to bed, but in short, multiple systems were in use. We have lots of weights and even some scales from archaeological digs at the Viking Age market town of Birka and other places, and there appear to have been variations in standards between places (as there were until the Metric System really). Already the earliest preserved town/marketplace laws in Scandinavia (Bjarkey laws) establish that local authorities can standardize weights and measures. Even if those are post-Viking Age, there's no reason to assume they would've been much different earlier. Anyway, for silver, the weight units in use around the Baltic seem to have been based off the Arabic silver dirham (just under 8g) and mithqal (~4.5g) units from about the 9th century forward.

Erik Sperber was the authority on this stuff (I have to mention he was an amazing guy. Was a PhD in chemistry who worked as a pharma researcher, then started a second career in archaeology after retirement, getting his second doctorate at the age of 85, a world record at the time, and continued to work with that full time until he was 95!). Anyway, to quote at length the summary of his article Balances and weights in Viking Age Sweden (pdf file):

The small, 0.7-4.5 gram, cubo-octaedric weights found at Bandlunde and Paviken, Gotland, Birka, Upland and Hedeby, Germany, belong to the same system. It is probably based upon the weight dirhem = 2.83g although the canonical dirhem = 2.97g cannot be entirely excluded. The bronze clad iron weights were estimated indirectly by measuring and calculating their volume which was then converted into the weight. The Bandlunde weights belong to a system probably based upon the mitqal sanctioned by the chalif Abd al Malik in Bagdad AD 696.Their unit is c 4.25g.The Birka unit was only 4.0g. Some solid bronze weights derive from a unit 3.0 g. It is an impossible task to map all the different weighing systems used during the Medieval Age simply because administration in most areas was weak and did not own the technical facilities needed for a rigid control of weights and balances. For one type of trade, only, one may expect international weighing systems, the trade with precious metals. The high and persistent value of the goods made the weighing meaningful. A piece of silver kept its weight even if it was carried to another market. In Scandinavia there have been found more than one hundred balances and several hundred weight pieces from the Viking age, mainly from the tenth and eleventh century. They are remarkably uniform and several authors have studied them and tried to attribute them to the various weight system prevailing in Europe and the near East. The problem is, however that these systems are rather similar and that the exact nominal weight of the often severely corroded pieces is difficult and frequently impossible to establish.

The full paper's in the link. Another, older, Sperber article is "How accurate was Viking Age weighing" (pdf), and he wrote a whole book: Balances, weights and weighing in ancient and early medieval Sweden , Stockholm, 1996

More recent results show that two different mitqal units may have been used at Birka, different for exports and imports, as a way of taxing trade.