Nuluk-khizdîn is the name the Petty-dwarves gave to the original caves of Nargothrond when it was their home. At some point they lost it, and eventually Finrod Felagund claimed them and founded the realm of Nargothrond. He expanded the caverns, gaining the Khuzdul moniker Felak-gundu in the process. No translation is ever given for Nuluk-khizdîn. The only clues we really have to it's meaning come from an old form that Tolkien discarded: Nulukhizidûn.
Nuluk then was originally nulu, as the final <k> was added later. The radicals for nulu match that of nâla from Kibil-nâla, which we know to be "waterway, channel, path, course". The root would be N-L-', so we have to ask where the <k> enters. That would apparently be due to the presence of <kh> at the start of Khizdîn. The glottal stop < ' > isn't very far away from the velar stop <k>. I could definitely see the glottal stop being assimilated to <k>. This would be the only occasion we see this in Khuzdul, but in no other words do a < ' > and <k>, <kh>, or <g> come into contact. I can especially see this occuring if the final syllable in nulu(k) was long and therefore stressed. From this we can guess that Khuzdul has nâla "waterway", singular, and probably (?) nulû "waterways", plural. This interpretation of nulu(k) makes a lot of sense. One of the defining features of the caverns is that they are located where the rivers Narog and Ringwil meet.
Nulu(k) then is "waterway, channel, path, course" and is plural, nominative, composition.
Khizdîn is a little more difficult to decipher. It has a very similar structure to Tharkûn and Nargûn, but with a different vocalization. The old version of the name, Nulukhizidûn, shows a -ûn suffix like Tharkûn and Nargûn, so it seems that Khizdîn most likely shares a similar etymology. The suffix -în is probably also a "suffix of specificity", just like those other words. With the Kh-Z-D root, Khizdîn would then mean "Dwarf one", or "Dwarf place", depending on the translation you prefer. The -în ending isn't too puzzling, as we see different vowels in Tharkûn, Nargûn, and Gabilân. In Arabic, different vowels sometimes show up in what is considered to be the same affix. It could potentially be a difference of gender as well. In Arabic, the names of towns, cities, and countries are usually feminine, and in Adunaic feminine names are formed with the suffixes s -i(-), -î(-), and -ê(-). Thus -în may simply be the feminine form of the suffixes -ûn and -ân.
The next question to decipher is why the vocalization has Khizd- instead of Khuzd. With Tharkûn, Nargûn, and Gabilân, all three can be viewed as coming from composition forms that we see elsewhere in the existing Khuzdul corpus. Khizd-, however, is the only place we see this. Had Tolkien decided on calling the place Nuluk-khuzdîn, it would be case closed. Because Nuluk-khizdîn is associated with the Petty-dwarves, the only thing I can think of is that there is a different word using the Kh-Z-D root that means "Petty-dwarf / -dwarves". Arabic has a template for diminutives, CuCayC, that is very productive. I don't see that exact pattern being in use here, but perhaps Khuzdul has a diminutive that results in the vowel change we see here. A diminutive form of Khuzd "Dwarf" would certainly fit the concept of "Petty-dwarf" perfectly. The Petty-dwarves are, if nothing else, a diminutive class of Dwarves, at least in spirit and attitude. They seem to be a sort of "untouchables" class of old Dwarven society. Mîm the Petty-dwarf explained to Turin that his people were Petty-dwarves, so it seems that they may have indeed referred to themselves as such. Nuluk-khizdîn was a home for the Petty-dwarves, so the Khizd- stem may be part of a plural form, in particular one that had a composition form of CiCiC that we see in the old form Khizidûn. Perhaps the pattern CiCiC, by virtue of having two of the same vowel, would also have it's second vowel suppressed with the -ûn/-ân/-în suffix, just like CaCaC in narag → Nargûn.
Khizdîn, from what evidence there is, appears to be "Petty-dwarves one/place", and would be singular, nominative, indefinite.
Nuluk-khizdîn as a whole would be "Petty-dwarves-place of the Rivers" and is a compound word with noun-noun word order.
The Silmarillion, ch 22, pg 231
The Silmarillion, index, pg 344
The War of the Jewels, pg 180