I would like to ask how to interpret the "もの" or "もん" in the phrase "どうしたものでしょうか" or "どうしたもんか". Does it have any specific meaning or serve a grammatical function? I know the phrases have the same meaning as "どうすればいい?", but I am curious about this もの, though.
ものだ after a verb has several special functions, and one of them is to describe an advice based on common sense.
So どうしたものでしょうか is like どうするべきでしょうか. Basically it's seeking for what an ordinary person would do in a similar situation. That said, どうしたものか/どうしたものでしょうか has been established as a set phrase said when someone is at a loss, and you may not have to take it as a genuine question and answer it.