How did 綿蠻 come to mean "(the chirping of) a small bird"?
From Shijing:


  《詩・小雅・綿蠻》
  
  綿蠻黃鳥，止於丘阿。道之雲遠，我勞如何。飲之食之，教之誨之。命彼後車，謂之載之。
  
  綿蠻黃鳥，止於丘隅。豈敢憚行，畏不能趨。飲之食之，教之誨之。命彼後車，謂之載之。
  
  綿蠻黃鳥，止於丘側。豈敢憚行，畏不能極。飲之食之，教之誨之。命彼後車，謂之載之。


Firstly, some conceptual advice on interpreting characters: Please keep in mind that characters are not words; they only represent words. Character phonology is very important when interpreting pre-Qin texts, of which the Shijing is one.

You should always be mindful that a character has both meaning and sound, and sometimes, characters are only used for sound.


  If you're a Japanese learner, you should be aware of ateji, which is based on the same principle.




Character definitions


  《說文》：“緜（綿），聯微也。从糸，从帛。”


「綿」(Zhengzhang OC: /*men/, thin, fine threads) is comprised of semantic「糸」(silk threads) and semantic「帛」(silk/fabric).


  《說文》：“蠻，南蠻，蛇穜。从虫，䜌聲。”


「蠻」(/*mroːn/, Southern ethnic tribes) is comprised of semantic「虫」(poisonous snake, now written「虺」, the cultural symbol of some ancient Southeastern Tribes) and phonetic「䜌」(/*b·roːn/).「閩」( /*mrɯn/, /*mɯn/, Min ethnicity), with the same semantic「虫」, is almost certainly cognate to「蠻」.

Stringing the above together,  you get a reading which is approximately /*men mroːn/, where「蠻」is almost definitely only used to represent sound.



How did 綿蠻 /*men mroːn/ come to mean “(the chirping of) a small bird”?

Two interpretations, which may be related to each other:


綿蠻 (/*men mroːn/) is short for 綿綿蠻蠻, where


綿綿 means fine, delicate.


  《淮南子・繆稱訓》：“福之萌也綿綿，禍之生也分分，福禍之始萌微，故民嫚之。”

蠻蠻 (/*mroːn mroːn/) is the topolectical name of a bird.


  《山海經・西次三經》之首，曰崇吾之山，在河之南，北望冡遂，南望䍃之澤，西望帝之搏、獸之丘，東望䗡淵。有木焉，員葉而白柎，赤華而黑理，其實如枳，食之宜子孫。有獸焉，其狀如禺而文臂，豹虎而善投，名曰舉父。有鳥焉，其狀如鳧，而一翼一目，相得乃飛，名曰蠻蠻，見則天下大水。



Together, this forms delicate > small + bird, i.e. small bird, extended to mean chirps of a small bird.
綿蠻 (/*men mroːn/) is a phonologically-shifted reduplication of「微」(微微, /*mɯl mɯl/).「微」means tiny, slight.


  王國維《觀堂集林・爾雅草木蟲魚鳥獸名釋例下》：“蟲之小者曰蠛蠓 (/*meːd  moːŋ|moːŋʔ/)，鳥之小者亦曰綿蠻，殆皆微字之音轉。”
  
  Small insects are called 蠛蠓 (/*meːd  moːŋ|moːŋʔ/), small birds are called 綿蠻 (/*men mroːn/), they are both phonological shifts from「微」(/*mɯl/).


Contextually, the reduplicated word meaning small was only used to describe birds, hence it became associated with birds to get small birds > chirping of small birds.




References:


《王力古漢語字典》
《漢語大詞典》