thick and thin

The girls just came out from animated discussion to ask me:
In the phrase "through thick and thin" which is the better one?
Elder daughter was convinced that they refer to the amount of resources, so thick is better.
Younger argued the opposite.
I asked whether they would rather go through thick brambles or thin ones?

On the first page of google's results the interpretations go both ways.

Is the glass half full, or half empty?

Disregarding the argument, they're coming with me to Victorian England on Sunday, so I win.