The emphasis is on the first syllable :-)

Malini is an Indian name and probably not familiar to most non-Indians...
Here are various meanings of Malini from the Monier Monier-Williams Sanskrit English Dictionary (Big thank you to Parul's friend Malini Dhondiyal for this information!)

Female florist/wife of a garland-maker

Music - A particular Sruti, Samgit ; Name of various metres

Name of Durga (wife of Shiva, mother of Ganesh)

Name assumed by Draupadi during Agyatvas

Name of the celestial Ganges

Name of a city in the Mahabharata

We liked the sound of the name. Mira can already say it very well...