יהוה, or Yah, a shortened form of the name. It dates back to an old Jewish custom. Because Jews didn't want to blaspheme whenever they came across the covenant Name of God in Scripture they pronounced the word Adonai instead, the Hebrew word for "Lord." I believe Christianity should discontinue this practice. While noble it wasn't a Hebrew practice from the beginning; it developed later in Jewish history.
God's covenant Name is important to him and it is used extensively in the Old Testament. In Hebrew thought a person's name reveals his character, his essence. God revealed himself as YHWH, the Hebrew name for the Trinity of Father, Son and Holy Spirit. It comes to us from the Old Testament and means, "He Is"—that is, God's presence is with his people and, thus, all his blessings that come with him. (cf. Exodus 3:15).
Written Hebrew didn't provide vowels, simply consonants (Y H W H, sometimes called The Tetragrammaton, The Four Letters), so there is uncertainty in how it should be pronounced. The American Standard Version (1901) is admirable in its attempt to restore God's Name but it used Jehovah to do so. (The "J" is to be pronounced as "Y".) Modern scholarship is generally agreed on Yahweh (pronounced as yau'whay) but, of course, we have no audio recordings from ancient times to substantiate the view.
I believe we should honor the name God of great covenants with people and substitute LORD from the Old Testament versions with Yahweh. If some are sensitive about getting Yahweh wrong—as the ASV got Jehovah wrong—then the printed Bibles could spell it YHWH or YaHWeH.
Restore God's covenant Name to the pages of the Old Testament!
One warning—don't treat the name Yahweh as magical. Years ago I came across a guy on the Internet. As memory serves he quoted Psalms 81. 14b, "I will protect him, because he knows my name" (ESV). The man treated Yahweh like a magical incantation, a verbal talisman or charm which, if you use it, God provides special blessings that he wouldn't give others who don't. I strongly reject such thinking. The Religion of Yahweh—the covenants governing their times—is not Magick. Jewish exorcists of old were sought after because it was believed they knew how to pronounce Yahweh properly. Such superstition and Occult dealings have no place in the worship of Yahweh.
UPDATE: Since I wrote this post I became aware of the New Heart English Bible (YHWH Edition), a public domain translation, that has restored Yahweh in the place of LORD. You can download it for free here.