What is the significance of the double Yud used as a stand-in for G-d's name?
When you have two Yudin (or in Polish Jewish pronunciation, "Yidin" - also means "Jews") who get together on an equal level, neither of them putting themselves higher than the other, then you have Hashem's presence there. (The "Holy Jew" of Pshischa, cited in R' S.Y. Zevin's Sippurei Chassidim to Num. 14:20)
Rav Tzadok Hakohen from Lublin answers in his Sefer Takanas Hashavim that it may be the first Yud represents HAshems name the Tetragrammaton (The Shem Havayah). The second Yud is from the last letter of the name of Adnus.
From what I'd heard (YUtorah shiur on sheimos), the original version was a double-yud with a horizontal line on-top, which was supposed to look like a "vav." So yud + yud + vav = 26, same count as Yud-then-Heh-then-Vav-then-Heh, it was a non-sheimos way of indicating the Name.