According to the character sheet, Melody has 82 skill points. But at level 9, with +6 int bonus, being a wizard, and taking +1sp per level for favoured class, it should be (2+6+1)*9 = 81. Where's the extra skill point coming from?

It is coming from your Headband of Vast Intelligence +2. "A headband of vast intelligence has one skill associated with it per +2 bonus it grants."

The skill points granted by Headband of intellect are in correlation with the intelligence bonus provided. So 81 would be correct. Manual bonus should be removed