How might a Jehovah's Witness support its usage of "Jehovah" over all names for God, and why does the name "Jehovah" convey the desired information?
The understanding of the meaning of the word name (hebr. shem / gr. onoma) in general tends to be too narrow in the context of the modern use. In the ancient languages the word name meant the person. The name was semantically not a referent, at least not primarily.
Therefore the Name of YHWH is not the Tetragrammaton (neither it is Jehovah, nor Jahweh). The Name is the One Himself. The letters to the word have a meaning, but this is not the Name. The meaning was almost a chiffre: He Will Be. It was a promise. It may have been a threat to some. He did not reveal himself to Moses (through the angel) saying (in regard to the people): I shall be your Father. He said: hjh 'schr hjh. I shall be as I shall be.
Were they (scribes? priests?) right to speak and even write Adhonai and Elohim (and Kyrios and Theos) when they read these letters?
They may have been wrong in their reasons. Still they may have been right in the result: The revelation of God did not go backwards. It advanced.
Reintroducing the Letters after the coming of the Son of God would be like saying: Moses is more than the Teacher that would come after him. Going back to Moses' revelation would be like saying: There is no other to expect.
Introducing the transliterated Tetragrammaton with medieval vocalization to the NT texts is a decision that reveals a thorough disregard for the impact of the coming of Christ to the revelation of God. It is an attempt (as an overreaction to Trinitarian dogma) to separate Father from Son, to override the new way of speaking of God and Christ as shown by the Apostles in their letters.
I can appreciate some challenging element in this translational approach. With other translations it has in common that not every decision has been thoroughly thought through.
The question asks for a positive justification for the use of that specific Latin-alphabet rendering. I don't see where a text-based justification is going to come from. That rendering is the result of a misunderstanding. In traditional Jewish pointed orthography, the four consonants of the name are written with those vowels -- not because anyone anywhere in the Masoretic tradition ever thought that the word was pronounced that way, but to serve as a reminder not to try for the original pronunciation. See http://hermeneutics.stackexchange.com/a/6063/947 and surrounding answers for plenty of detail. Later, non-Jewish readers mistook this for the actual pronunciation, and we were off for the races.
In making a translation, some think that it's important to make it clear where the different Hebrew deity-words appear. If some translator into English decided to uniformly use Jehovah for the four-letter name (where others choose LORD, or others put the Hebrew letters), and use other words for the other words, I suppose that's better than just picking words at random.
In the Hebrew Scriptures
The Tetragrammaton (four-letter name of God) appears multiple times in the Hebrew Scriptures. Many translations render this as LORD, following the Jewish practice of not pronouncing the Divine Name (though the Jews do write the name in their scriptures). The Jerusalem Bible renders the name as Yahweh, which is a scholarly “best guess” at the original pronunciation. The New World Translation uses Jehovah, which is almost certainly not the original pronunciation, but is the traditional rendering in English, found in both religious and secular books for many many years. Certainly including some form of the name is more accurate than bowdlerising it.
So, the name is there, in the text, and the translation includes it. Translations which leave it out are the ones which need to explain themselves, not the ones which leave it in.
In the Christian Greek Scriptures
The New World Translation also uses the name Jehovah in the Greek Scriptures, although it is not found there in any extant manuscript. When the Greek text quotes the Septuagint, they reinsert the name (yes, reinsert, as they maintain that it was there originally). Certainly there do exist editions of the Septuagint which contain the untranslated and untransliterated Tetragrammaton, and others which render the divine name as Pipi, suggesting that they were copied from an earlier version which contained the original Tetragrammaton, the Hebrew letters of which look a little like the Greek letters for Pipi. So, where the Christian scriptures quote passages from the Septuagint, it is somewhat reasonable to assume that the Tetragrammaton, or some variant thereof, existed in the original text and was later lost.
However, NWT includes the divine name in other places too. Sometimes support comes from the existence of the Tetragrammaton in Hebrew translations of the Christian Greek Scriptures (some of those Hebrew translations used for support are actually fairly recent, so any support they offer is tenuous at best). The name Jehovah occurs many times in the New World Translation of the Christian Greek Scriptures, each time accompanied by a footnote and an explanation of the rationale in The New World Translation — with References.