What is meaning of the word 'Kamakshi'? What does the word 'Kama' here refers to ?
Parvati Devi Herself describes the meaning of Kamakshi in Chapter 4 Purvaardha - Arunachala Mahatmya of Skanda Mahapurana.
Goddess Parvati once playfully closed eyes of Lord Shiva and entire world went dark creating chaos everywhere. Parvati Devi felt bad for Her immature act and went to Kanchi and did tapasya by establishing Ekambareswarara Linga (under a mango tree and hence the name Ekambara). She Herself manifested as Kamakshi which means who fulfills all their desires.
- I am called Kåmäkshi on account of my bestowing every thing desired. Let my devotees here obtain the desired boon after bowing down to me under thc name Kämäkshi as they please.
Word Kama (काम) denotes wish or desire.
As far as I know, Kamakshi is considered a form of the Devi. Lalita Sahasranamam also mentions Kamakshi as one of the 1000 names of the Devi.
The word Kamah (कामः) has the following meanings
object of desire
love or desire of sensual enjoyments, considered as one of the ends of life
desire of carnal gratification, lust
the god of love
a name of Pradyumna
a name of Balarama
a kind of mango tree
the Supreme Being
The word Akshi (अक्षि) means
the eye (which grasps or sees objects)
the number two
The most common meaning of the word Kamah used here is in the context of desire. Hence, Kamakshi can be taken to mean 'one whose sight fulfills all desires'. In my opinion, this can be interpreted in the context of Darshana. It is said, when the Guru looks at you directly (eye-contact), that is considered very auspicious.
I imagine it could also be interpreted as one whose eyes are full of love.
Reference used: The Practical Sanskrit-English Dictionary by Vaman Shivaram Apte.