I'm not sure what the "official" word on this subject is, but when I was with a small arts organization the gift shop issue fell to me as well. I was told to figure out how to make it more profitable.

Using some simple math with numbers I got from the Financial director and my retail background, I was able to calculate that the amount of space we were using to display items for sale and the amount of time we were spending keeping track of approx. 2,000 items was not justified. Very little inventory was moving and we were just counting it over and over...a lot of new competition from nearby retailers ended our once lucrative gift shop business several years earlier.

So I recommended that we close the gift store keeping only the three greeting card towers because those were generating approx. 80% of the income. We then used the space to rent out small sections to artists who would be juried and then they could display their original art for sale. Because they rented the space (an immediate increase on $ generated per sq. ft. by around 5x) we didn't keep any commission. In our area there was a need for amateur/emerging artists to have access to an affordable way to sell their work. We handled the sales and collected tax and sent checks to artists each quarter.

Additionally, selling cheap trinkets made by Two's Company and other manufacturers had nothing to do with us as an organization, selling original artwork was perfectly aligned with our mission as an organization. 

The entire process was an easy explanation to the board.

I'm not saying that closing your museum shop is the way to go, but you have to be open to the possibility that it just isn't working the way it is and sometimes a major change is needed. Especially if you're having to justify expenses - 

Good luck!