Kanji "stone" contains (cliff 厂 + mouth 口).. my question is what is the idea of mixing mouth with cliff ... the cliff function here is clear because it's some thing related with stone.. but what is the mouth function?
which is a depiction of stone chimes. The original meaning of「石」is stone, rock.
There was an ancient musical instrument made out of stone chimes called「磬」.「磬」was originally「声」, which depicts stone chimes「石」suspended with threads/string (now looking like「士」as the top part of「声」). In the character「声」, the stone chimes component「石」is written as「𠃜」.
Later, a hand holding a striking implement「殳」was added onto「声」, forming「殸」, to emphasise that the instrument is meant to be struck. Finally, another semantic「石」(rock, stone) was added on to「殸」, as a further emphasis that the instrument was made of stone.
In modern Japanese,「声」is used to represent another word, which was originally written as「聲」(sound, extended to mean voice).「聲」(Baxter-Sagart OC: /*[l̥]eŋ/) is a compound of「殸」(/*[C.qʰ]ˤeŋ-s/, stone chimes being struck) and「耳」(ear), indicating the meaning sound.
A distinguishing symbol「口」was added to mark the name of an ancient region called「石」. This distinguishing symbol has been inherited in some variants of the character up until the modern form of「石」.
Distinguishing symbols were often added to existing characters to form new characters, and some of these symbols are inherited in the modern form. The symbol「口」or its derivatives「甘・曰」was added to a few older characters to form proper nouns (names of states/regions), such as「商」(Shang Dynasty),「周」(Zhou Dynasty),「曹」(State of Cao),「曾」(State of Zeng), etc, and their modern forms have permanently inherited this symbol (regardless of whether they're used to represent a proper noun or not).
The stone chimes component was later simplified into「厂」, then the strokes were further altered slightly into「丆」, resulting in the modern shape.
There are characters built from variants of「石」which may or may not have the distinguishing symbol「口」. In fact,「石」was subject to several different shape changes across different characters, obscuring their connection to「石」:
In some characters, a few strokes in the form of horizontal lines (probably decorative with no meaningful function) were added on to「石」, as seen in the second form of「声・殸・磬」.
These shapes are present (without「口」) in e.g.:
In some characters,「石」is written as「⿸广廿」, such as in「度」,「席」, and「庶」.