Azanulbizar is the name given in The Lord of the Rings to the valley that lies amid the mountains of Moria: Zirak-Zigil, Barazinbar, and Bundushathûr. It is here that Durin I saw the crown of seven stars above his daytime reflection in Kheled-zâram, the Mirrormere lake. The final battle in the War of Dwarves and Orcs was fought here. The East Gate of Moria opens onto Azanulbizar, and it saw the Fellowship flee from Moria after Gandalf's battle with the Balrog and thence into Lothlorien.
In The Treason of Isengard and Parma Eldalamberon XVII, we read the gloss that Tolkien wrote in his Words, Phrases and Passages. That is:
Azanul-bizar, uncertain, but probably 'ZN = dark, dim and ûl = streams? bizar, a dale or valley.
That is, according to this view, Azanulbizar consists of azan "dim", ul or ûl "rills, streams", and bizar "dale, vale, valley". To go along with this, Tolkien also proposed a form Azanûl, which would apparently be "Dim Streams" and be a shortened name for the valley, similar to what we see for Zirak-Zigil, Barazinbar, and Bundushathûr. This appears to have subsequently discarded.
However, Tolkien also wrote another analysis of the word, which is found in his manuscript entitled Nomenclature. This can be found in A Tolkien Compass, and now in an updated form in The Lord of the Rings: A Reader's Companion. As near as I've been able to tell, Words, Phrases and Passages was being written circa 1958-1960, and was a result of fans desiring translations and Rayner Unwin requesting information for an index to The Lord of the Rings. Nomenclature was Tolkien's effort to guide other writers' efforts in translating The Hobbit and The Lord of the Rings, for he was disappointed with the first such effort to translate them into Dutch. In 1967, Tolkien wrote a letter saying that he had almost completed work on the Nomenclature manuscript, which places it at a later date than Words, Phrases and Passages. From this, it appears to me that the explanation found in Nomenclature represents an update in Tolkien's thoughts on the matter. Also, the explanation given here syncs much better with the rest of the corpus than the etymology found in Words, Phrases and Passages. The result is that I will work with this explanation, which uses the phrase Duban Azanulbizar, rather than trying to cram the entirety of "Valley of the Dim Rills" into only Azanulbizar.
Duban is "valley", according to Tolkien's Nomenclature, with root D-B-N. As the first element of the phrase, which is apparently a construct phrase, it should be in composition form. Tolkien doesn't state it explicitly, but duban is apparently also singular, and I would say nominative. It's interesting that it shows up, for its form CuCaC has the same relationship to the vocalization of uzbad, uCCaC, as does zirak CiCaC to inbar iCCaC.
Duban is thus "valley" and most likely singular, nominative, composition.
In Nomenclature, Azanul is azan "shadows" plus -ul, which Tolkien calls a "genitive ending of patrynomics...". This is the same ending seen in Fundinul. He says azan is a plural of uzn "dimness, shadow", so the indefinite plural is probably azân, with composition form azan, parallel to Khuzd, Khazâd.
The -ul seems to be more of an adjectival meaning than true genitive. In Khuzdul "Dwarvish" it corresponds to the English suffix -ish, whereas in Azanul it seems to correlate with English -y or -ed, such that "shadows" becomes "shadowy". Arabic has a direct parallel, the nisba, which is formed with the suffix -iyy, or -iyyât in the feminine. The nisba is used to form adjectives from nouns, just like we see here. Due to the structure of Arabic, the resulting adjective can also be used as a noun, which is what we see with the word Khuzdul. It seems pretty safe to assume that -ul is Khuzdul's "nisba". It's meaning would be "a thing associated with or related to the root word".
Azanul then should be "shadowy, shadowed" or and adjectival, plural, nominative, composition.
Bizar is "rills, streams". Sometimes Tolkien wrote this as bizâr, as he did in the Nomenclature gloss. CiCâC is a plural form found in Arabic, and here also Tolkien suggests that it is "probably a plural of a stem B-Z-R." Perhaps then bizâr is technically the plural, but sometimes the -â- gets shortened to -a-, so the pattern is CiCaC, which is a singular pattern also seen in both Arabic and Khuzdul. As such, it could almost be like a collective. In either instance, it would be nominative and indefinite as the second element of the compound.
I'm going to say bizâr is "rills, streams" and is plural, nominative, indefinite, while bizar is (informally) collective, nominative, indefinite.
The full compound Azanulbizar is "dim rills" or "shadowy streams", and is itself a compound word of adjective-noun word order. As a compound, it is nominative and indefinite since it is the second element in the full phrase.
The whole phrase Duban Azanulbizar is then a construct phrase, with a noun as the first element and an adjective-noun compound as the second.
The Lord of the Rings, The Fellowship of the Ring, Book II, ch 4
The Lord of the Rings: A Reader's Companion, pg 269
The Return of the Shadow, pg 465-466
Parma Eldalamberon XVII: Words, Phrases and Passages, pg 37
A Tolkien Compass, pg 182