The place-name Kinnegar may be an anglicised form of Ir. coinicéar/coinigéar, 'rabbit warren'. It is also possible that the name comes from English coneygar or Scots coneygair with the same meaning, but the vowel in the first syllable of Kinnegar suggests that an origin from the Irish word is more likely.
It is not clear which place Kinnegar Road is named after as there are several places with this name, including one near Holywood, Co. Down and another near Rathmullan, Co. Donegal. Perhaps the Holywood name is most likely as other streets nearby in Finaghy are mainly named after places in Co. Down and Co. Antrim, but there is also Inishowen Drive, named after the peninsula in Co. Donegal.
Note that there is also a Kinnegar Road in Holywood, close to Kinnegar, once an area of sand dunes and a rabbit warren. It is next to the townland of Knocknagoney, which also references rabbits, as it is an anglicisation of Ir. Cnoc na gCoiníní ‘hill of the rabbits'.