Myrtis was the name of an ancient Greek woman poet.
http://books.google.com/books?id=ncN7uneLKrcC&pg=PA892&lpg=PA892&dq=myrtis+poet&source=bl&ots=K4tWaCSfiC&sig=AA9aN0fbtwE8NE0XLU5V8mVNoTY&hl=en&ei=zo03ToXLFIWDgAeE--CzAg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBUQ6AEwAA#v=onepage&q=myrtis%20poet&f=false
Myrtis was known in the 19th century partly because of a poem written about her by the British poet
Walter Savage Landor:
http://www.poemhunter.com/poem/myrtis/
I haven't found whether or not her name was related to the ancient Greek word for "myrtle", which was
myrtos. In spite of the similarity between the two, you'd have to get confirmation from a scholar of ancient Greek names to know if the two words were truly related.
There are quite a few names that have been "brought back" from ancient Greek sources at different periods in history. Whether or not Myrtis is etymologically related to
Myrtle, Myrtis's use as a given name in the USA was probably inspired by its similarity to
Myrtle and the fact that it sounded like a blend between
Myrtle and
Alice. Another ancient Greek name which became much more popular in the USA than Myrtis was
Doris, which blends the sounds of
Dora or
Dorothy with the ending of
Alice.
The fact that your grandmother's parents were immigrants from Sweden may have little to do with their choice of the name. Many immigrants have consciously tried to give their children "American" names to help them fit in. But they often make mistakes because they aren't fully integrated into American culture yet. The spelling "Myrtez" used in your family may be a mistake partly caused by your great-grandparents' Swedish accent, though Myrtis certainly was often respelled as Myrtice, Mertice, or Mirtis by people who liked the sound but didn't know the spelling. Your great-grandparents probably knew or heard of some American woman with the name Myrtis and thought it would be a good one for their American-born daughter.