In the case of the Holy Trinity Monastery, Dreifaltigkeit can be recognized, as drei = three and keit more or less translates to ness. However as I thought about the compound, faltig had me stumped. How did the Trinity come to be described with the word "wrinkled"?? At least that's the only meaning I was familiar with. So I did some searching online which seemed to confirm the mundane, that faltig just means "wrinkled".
Then I came across the digitized version of A Grammar Of The German Language For High Schools And Colleges by Hermann Carl George Brandt (Allyn and Bacon, 1894), which gives faltig's archaic meaning of "folded" from the verb falten "to fold".
I do not know to what extent Dreifaltigkeit is used less often than Trinität or Dreieinigkeit since the elements were compounded in older times. But even using an archaic word, it is still translated as "Three-folded-ness". I find this a terribly beautiful and a poetic image.