It varies quite a bit, depending on the length, subspecies, and fat condition of the individual whale. I’ll give a short and long answer.
Short answer: Between 55 and 180 tons on average.
Long answer: Or, how I got the short answer.
There are three subspecies of blue whale. Pygmy, Northern, and Antarctic.
From my digging through scientific papers, I've found that, when fully grown, the average length for each subspecies is:
Pygmy: 68-74 feet
Northern: roughly 70-85 feet
Antarctic: roughly 75-90 feet
The confirmed record for each subspecies is:
Pygmy: 79 feet
Northern: Uncertain, probably about 90 feet
Antarctic: 98 feet
The paper "Body Weights of some Species of Large Whales" (Lockyer, 1976) gives formulas for the average weights of whales at given lengths, based on actual weighings. There are formulas for Antarctic and Pygmy blue whales, but not Northern, because few Northern blue whales were weighed. However, the few that were were more in line with the Antarctic formula than the Pygmy formula, so I will use the former for them. The original formulas did not include blood loss, (blue whales were only weighed by parts, since they are much too large to fit on any scale) but the paper gave 6% as the average weight lost from blood, so I have modified the formulas accordingly. I have also converted them into imperial units. (Sorry, metric users! have the imperial formulas memorized (because I'm obsessed with blue whales), I'll have to look up the metric ones again before I can post them here)
The formula for Antarctic blue whales is:
W = 0.0001368 • (L^3.09)
L = length in feet, W = weight in short tons.
For Pygmy blue whales the formula is:
W = 0.00000418 • (L^3.97)
These formulas only give the theoretical average weight of a blue whale of a given length. In reality the weight of individual blue whales even of the same length will vary substantially.
In the paper "Growth and Energy Budgets of Large Baleen Whales from the Southern Hemisphere" (Lockyer 1980) we learn that Antarctic blue whales usually gain 50% of their lean body weight during the summer feeding season. That means that a blue whale entering the Southern Ocean in the spring weighing 100 tons would leave weighing 150.
Therefore the fattened weight is 120% the average weight and the lean weight is 80% of average. Other blue whale subspecies are not as strict about their migrations and feeding habits as their Antarctic cousins, but they can still gain and lose a lot of weight, and their physiology is not too different.
Therefore we can assume that most blue whales will be between 80% and 120% of their predicted average weight.
Now all we need to do to find the typical weight range of fully grown blue whales is to multiply the average weight of the shortest end of the length scale by 0.8 and the longest by 1.2. This gives the average weight range for each subspecies as*:
Pygmy: 60-135 tons
Northern: 55-150 tons
Antarctic: 65-180 tons
The weight of the largest individuals would have been**:
Pygmy: 114-171.5 tons
Northern (assuming 90 feet): 119.5-179.5 tons
Antarctic: 155.5-233.5 tons
I'm glad I was finally able to use this, and I hope it helps! :-)
*These have been rounded to the nearest 5 tons. When off by more than a ton they have been rounded upward at the high end of the scale and downward at the low end.
*Rounded to the nearest 0.5 tons.