Why the weight changes
Ingredients do not have the same density. A cup of honey weighs much more than a cup of flour, because honey is denser. So convert ingredient by ingredient, never with a single number for everything.
The usual equivalents
These values are approximate, because the real weight varies with packing and brand: all-purpose flour, about 120 g per cup; granulated sugar, about 200 g; packed brown sugar, about 220 g; butter, about 227 g; cocoa powder, about 100 g; honey, about 340 g; milk, about 245 g; water, about 240 g; oats, about 90 g.
An example
2 cups of flour make about 240 g, which is 2 times 120 g. 1 cup of granulated sugar makes about 200 g. The same cup, two ingredients, two very different weights.
For precision
For precision baking, weigh with a scale. Cup conversions stay handy for adapting a recipe quickly, but they remain approximate.