McIlwee or McElwee or............... the name means "son of the yellow lad" a reference to skin tone.
According to Edward MacLysaght - the Surnames of Ireland ISBN 0 7165 2366 3q.v the name comes from the Donegal-Derry counties of Ulster..........it has mutated into McGilloway, McKelv(e)y, McCalvey McGilvie, Kilvey, Kilboy, Calvey & Calway.
These variants mostly all sound the same despite their spelling.....especially when the speaker had a broad accent and the clerk writing down his name spoke with a different accent and spelling had not been standardised and wrote it as heard to his ears. In gaelic/erseGillie (lad) has become IL and the WEE part comes from Bhuidhe (yellow). The gaelic bh & mh can sound as english V.
There is no significance in whether the M' part is written as MC or MAC.....they both sound the same, and some of the derived names have come from dropping the MC prefix. In my family the prefix Mc and Mac have been used interchangeably, but usually an individual will settle on one or other and use it exclusively, but companies who send out bills etc write it any way they choose. This can now cause problems........last week my bank asked for proof of identity for an account and the electric bill with my name on it had Mc whereas my passport had Mac and so they wanted another utility bill to confirm which I was.
Lastly by way of nothing.......in my family tree there appears a John McWee, a grocer, died about 1907in Govan Glasgow. I was told by a relative that he was born McIlwee but changed it to McWee. Why ? I don't know. There appears to be a large number of Mc(IL)Wees in Ayrshire and Galloway......this is probably a result of immigration from Ireland as this is the nearest landfall fom Ulster.