This appears to be a Scottish "constructed" name, according to correspondents.
The mother of one correspondent [BM] was of Scottish stock and formally baptized Elma. This was understood to be from the elision of family names Elizabeth and Margaret [or, more precisely in this case, the variants Bethia and Marguerite].
Confirmation was given by another correspondent [ED2] who had a pet name from a similar elision: Elizabeth and Mary. She also says that the name is common in Aberdeen and, while she points out that elma is Turkish for "apple", she does not claim there to be a link!
In addition to agreeing to the above elisions as its source, Chambers suggests (unconvincingly) that Elma might also be a short form or derivative of Wilhelmina.