I found some information about SS EDWARD PIERCE but nothing about Ellef Andersen specifically. Here is what I found.
EDWARD PIERCE was a collier (a ship designed to carry coal) that was constructed by the Newport News Shipbuilding Company, Newport News, Virginia. The ship was completed and placed in service as of November 6, 1914. Her original owner was Crowell & Thurlow Steamship Company and the ship was named after Edward Pierce, who was then the Vice President (later President) of Crowell & Thurlow. For at least part of her career EDWARD PIERCE carried coal between Norfolk, Virginia (the location then and now of a major coal terminal), to Boston, Providence, Rhode Island, Searsport, Maine, and other New England ports. She also undertook trans-Atlantic voyages between North American and Scandinavia, sometimes carrying bulk cargo other than coal, such as flour, grain, cotton and wool. On at least one occasion she traveled from New York to Portland, Oregon, and return. The ship was scrapped in 1949. See http://shipbuildinghistory.com/history/shipyards/2large/active/newportnews.htm and scroll to hull number 182.
I found a remarkable photograph of what may be the same ship after it sank in shallow water in Boston harbor following a collision with another ship in 1924. In the photograph the name of the ship appears clearly as “EDWARD PEIRCE.” The information accompanying the photograph attempts to address the different spelling of the name although it is not clear to me from the explanation whether this might have been a different ship with a different spelling of the name, although other details seem to match the ship I describe above. See https://www.flickr.com/photos/boston_public_library/6215575412/. Presumably the ship was salvaged and returned to service.
In a newspaper article in the Nashua Telegraph dated August 5, 1937, the crew of EDWARD PIERCE, then in Lynn, Massachusetts, threatened to go on strike over “better living conditions.” https://news.google.com/newspapers?nid=2209&dat=19370805&id=bx5AAAAAIBAJ&sjid=pKQMAAAAIBAJ&pg=1029,2507730&safe=on&hl=en.
And incredibly enough, I found this entry in a book entitled “The Liberty Ships of World War II”:
“On December 21, 1941, the AP [Associated Press] reported that the 29 officers and sailors of Mystic’s [Mystic Steamship Company] ‘grimy coastwise collier’ EDWARD PIERCE, at Boston under Captain Ellenwood Folger, donated $124.61 to the Navy to ‘buy a bomber – a Xmas present to the U.S. Navy.’ It was accepted by Cdr. Robert Maborn and was believed to be the first such donation in the nation.” See https://books.google.com/books?id=A5oWBAAAQBAJ&pg=PA266&lpg=PA266&dq=%22edward+pierce%22+collier+-%22november+24%22+-oats&source=bl&ots=sYsfzxwkUv&sig=Btr04faJdChdstAeyWslTm-hiXg&hl=en&sa=X&ei=lRwtVZmRL8moNqfBgTg&ved=0CDQQ6AEwBw#v=onepage&q=%22edward%20pierce%22%20collier%20-%22november%2024%22%20-oats&f=false.
So if the above is correct, on December 22, 1941, EDWARD PIERCE was either in Boston or no more than one day’s sailing time away from Boston.
And if Ellef Andersen was injured in December 1941 (and died that month – from the same injury?), it may well have been caused by a shipboard accident.
If you want to pursue additional information about your great-uncle, although probably nothing more about EDWARD PIERCE, you may be able to obtain a copy of your great-uncle’s merchant marine career, by contacting the U.S. Coast Guard. (You say that you have his discharge but a discharge was issued at the end of each voyage so there is probably more to his record.) See this web page from the website I manage: http://armed-guard.com/searchmil.html. In particular see section A.2. Records of Individuals – Merchant Marine. You will have to contact the U.S. Coast Guard’s National Maritime Center in Martinsburg, West Virginia. The Coast Guard was and is responsible for issuing certain documents ("seaman's papers") to U.S. merchant mariners, so they may have information about your great-uncle. You will need to provide as much identifying information as possible about your great-uncle.
Note that you are not next of kin to your great-uncle (next of kin = parent, spouse, child, sibling). As such you personally may not be able to obtain his complete service record. It would be best if someone else in your family who is next of kin could make the request for his service record, for example a child of your great-uncle. You could do the leg work of preparing the request, researching information, etc., but it would be better if someone who is next of kin to your great-uncle could actually sign any necessary document(s). If there is no next of kin, make the request yourself and hope for the best. There may be a fee for this service but I expect the Coast Guard would not begin work without informing you of any charges.
Ron Carlson, Webmaster
Armed Guard / Merchant Marine website