I always thought the yud-yud was a random substitute for Shem Hashem. Until my Belzer chevrusa this Shabbos pointed to the Kitzur Shlah's statement that it represents the shiluv of the Shem Havayah and the Shem Adnus. (First letter of Havayah, last letter of Adnus.) I found this post that cites most of that explanation in the name of R' Aharon Feldman.
But if so, it is not so simple to say it has no kedushah. We know the Rogatchover held that even a Heh as a sign for Shem Hashem has kedushah, which is why he advocated the use of a Dalet instead. צ"ע.
See the fascinating Mishnas Chachamim below, courtesy of Reb Micha Berger, from
http://www.hebrewbooks.org/pdfpager.aspx?req=19594&st=&pgnum=9 on both the double-yud and the triple-yud:
Also courtesy of RMB, from Siddur Rav Sa'ada Gaon, the triple-yud "in action:"