Tony Hopkinson, Tauranga.
The place name Lynn (as opposed to a person's given name) is derived either from the old Celtic word lenna, meaning pool or lake, or, if located in Scotland, from the Scots Gaelic word meaning a water feature or pool.
However, interesting as this may be, it is not why New Lynn is so named.
Some time after European colonisation of the Auckland Isthmus began, the area of New Lynn was surveyed by Frederick Utting in 1863, who named it after King's Lynn in Norfolk because he thought the countryside was similar.
And as you surmise, Grey Lynn was indeed named after Governor Sir George Grey, who was variously Governor of South Australia, twice Governor of New Zealand, Governor of Cape Colony (South Africa), Premier of New Zealand and, towards the end of his life, the Member of Parliament for Auckland West.
The Lynn part of the name may have come from the fact that the area was very wet, and had to be drained before construction of dwellings could begin.