is the name the Petty-dwarves gave to the hill
Amon Rûdh, which translates as "Bald Hill" in Sindarin The top of the hill had a crown of exposed rock, which lead to the name.  For some time it was home to Mîm and his sons, and then Turin and his band as well.  A translation is not provided for the name, so most people guess that "Bald Hill" is probable.  It was common practice in Middle-earth to adopt names from other languages by either translating to words with the same meaning (such as Gabil-gathol Belegost) or simply render at least some of the words directly into phonological structures appropriate to the language (as in Khazad-dûm Hadhodrond).  Mîm was upset that "the Elves changed all the names", implying that the Petty-dwarves had probably inhabited the area before the Elves arrived.  The Elves originally hunted them down, but stopped once they had contact with the Dwarves of Gabil-gathol and Tumunzahar.  The original persecution seems to make it unlikely that they would have adopted the Khuzdul name of the hill into their language, but perhaps they didn't do so until well after they stopped attacking the Petty-dwarves.  Mîm's anger at the names being changed may have simply been his inability (or unwillingness) to understand that the Elves had simply translated the Khuzdul name, not actually changed it.  This does seem in accordance with the Petty-dwarves' view of the world.  To my knowledge, there's no other leads for translation, so I'll assume that "Bald Hill" is the correct interpretation.

The structure of the word Sharbhund is different from other Khuzdul words.  The middle cluster of consonants, <rbh>, makes it hard to decipher how to break it apart.  Going with <r> + <bh> doesn't seem likely, since Khuzdul doesn't tolerate initial consonant clusters, and it's highly doubtful that <bh> is an "aspirated" voiced bilabial stop.  Having such a phoneme would make Khuzdul's phonology much different than Adunaic, Arabic, or Hebrew.  A far more likely split is sharb + hund, which are syllables seen elsewhere in our Khuzdul examples and thus can be analyzed better.

Assuming that sharb is "bald", then it would be our only instance of an adjective showing the pattern CaCC, as opposed to CaCaC.  In Arabic, the root of "bald" is a stative verb, and it seems likely that Khuzdul would have the same.  Sharab could hypothetically be that stative verb "to be/become bald", along with a verbal adjective sharîb "bald", similar to gabil and gamil.  However, if the verbal adjective is used in this compound, the <i> doesn't seem like it would reduce when attached to hund, similar to how the <i> is retained in Gabilân.  Instead, the verb may have a different vocalization, and sharab may be the verbal adjective form.   Still, that would require a phonological rule that would reduce the final <a> due to the presence of hund, and I'm not wild about that idea.

Instead, sharb may be the gerund (verbal noun) form.  CaCC is found in Arabic gerunds.  If so, then sharb would be a noun meaning "balding; the process of becoming bald".  In Arabic, gerunds often have a  concrete meaning as well as conveying the act or process of the verb.  Here, we may have sharb also meaning "a balding; a place that has become bald or clear; a clearing".  We might simply call it "a bald spot".  Sharbhund would then mean "the bald area/spot hill" or "the hill of the bald area/spot".  This makes a lot of sense, given the description of Amon Rûdh.  It had birch, rowan, and thorn-trees growing on it's lower slopes.  The upper slopes were extremely steep and bare rock, and the top was covered only in red seregon (similar to a plant known today as "stonecrop").  So, it wasn't the whole of the hill that was bald, only the upper slopes.  Interpreting sharb as the gerund allows for a verb form of sharab and verbal adjective sharîb.  If the verbal adjective was used, as in Sharibhund, perhaps that would imply that the hill was in the process of becoming bald, or that the entire hill was bald and/or barren, rather than simply having a patch on top that was clear.

If sharb is "bald", then hund is "hill".  This is the same template as Khuzd, Khazâd, and it makes sense that it is singular, nominative, and indefinite.  The question is still how we have the cluster of three consonants in the middle of the word.  We should see a form more like Sharbahund. However, there are several consonants that are generally considered "weak" in Hebrew and Arabic: <'>, <y>, <w>, and <h>. These often lead to special word forms, especially when they are the second radical in a verbal root. Khuzdul seems to almost completely elide the glottal stop <'> when in the middle of compounds, such as Gundabad.  My opinion is that the weak consonants <'>, <y>, <w>, and <h> may become evanescent when they start a syllable that follows a closed syllable, as they do here. They become almost completely silent, and so the consonant cluster of -rbh- is allowed to stand as is.  No epenthetic vowel or other alteration is made.  If this is the case, the word is almost pronounced Shar-bund.  The weak consonants may, at most, add a bit of a secondary articulation to the preceding consonant, such that <h> would hint at aspiration (or "breathy voice" for voiced consonants) and <w> would lean towards a labialized consonant.  These would not be phonemic, and could be dropped entirely.  However, in the writing, the original weak consonant is still retained, except for the glottal stop apparently.

If this idea is correct, then there is a little bit more (very!) circumstantial evidence to support it.  If the original meaning is "Bald Hill" and the pronunciation becomes close to Shar-bund, then it becomes something of a play on words, since bund is "head".  Speakers would, essentially, be called Amon
Rûdh "Bald Head".  Perhaps this small bit of humor illustrates the fondness the Petty-dwarves had for the hill and why Mîm was so upset that the name changed at all, for this double-meaning is not present in the name Amon Rûdh.

Sharb is probably "balding, clearing" and/or "a bald/clear spot; a clearing" and singular, nominative, composition.

Hund is "hill" and singular, nominative, indefinite.

Sharbhund is thus "Bald-area Hill" and a compound word with adjective-noun word order.

The Unfinished Tales, pg 104