A possible source for this idea might be a story from the Bible, 2 Kings, Chapter 4. Quoting from the New International Version:
The wife of a man from the company of the prophets cried out to Elisha, “Your servant my husband is dead, and you know that he revered the Lord. But now his creditor is coming to take my two boys as his slaves.” [...]
Elisha said, “Go around and ask all your neighbors for empty jars. Don’t ask for just a few. Then go inside and shut the door behind you and your sons. Pour oil into all the jars, and as each is filled, put it to one side.”
She left him and shut the door behind her and her sons. They brought the jars to her and she kept pouring. When all the jars were full, she said to her son, “Bring me another one.”
But he replied, “There is not a jar left.” Then the oil stopped flowing.
She went and told the man of God, and he said, “Go, sell the oil and pay your debts. You and your sons can live on what is left.”
Here a woman collects seemingly valueless objects (empty jars from her neighbours) and yet they turn out to be her salvation, thanks to a miracle. Given that Father Brown is a Catholic priest, making references to a story from the Bible seems to fit well.
I found this by doing a Google search for do you know the story of the "woman who collected". Many of the search results were actually Father Brown stories, but one that wasn't was this page, which led me to the Bible story.