I always have some ash left in the barbecue. Would mixing it in the potting soil be a good or a bad thing for my pot plants? The ash is only from charcoal so nothing nasty in it.
It will definitely make a difference if you add enough, but probably not a useful one.
Ash is high in potash (compounds of potassium) and usually also high in calcium salts. This invariably raises the pH of soil. Ash unfortunately does not contain nitrogen, which is a more important plant nutrient than potassium.
Given that potassium is the third major plant and crop nutrient after nitrogen and phosphorus, ash is an excellent fertiliser in cases where you have a potassium deficiency and/or soil that is too acidic.
I wouldn't use it in potting mix because usually potting mix, by definition, already contains optimal nutrients to nurture young plants. I would consider using the ash for other purposes in the garden. There are several, given the key properties of ash, but I won't go into that here.