According to Michael's burial details his mother's maiden surname was Mulrane. He is interred in Cambusnethan cemetery, section E plot number 931. Here are the details.
MICHAEL REILLY, miner, Waterloo, aged 48 years. married.
Parents - Michael Reilly and Mary Mulrane.
interred in E931 on 11th February 1897.
Also in the same grave are -
MARGARET QUIGLEY, Waterloo, aged 11 months.
Parents - Peter Quigley and Mary Reilly.
Interred in E931 on 11th August 1897.
MARTHA QUIGLEY, Waterloo, aged 15 months.
Parents - Peter Quigley and Mary Reilly.
Interred in E931 on 3rd August 1899.
MICHAEL REILLY, Wishaw, aged 16 days.
Parents - Hugh Reilly and Sarah Burns.
Interred in E931 on 8th May 1908.
MARY ANN QUIGLEY, Motherwell Hospital, usual residence Waterloo, aged 4 years & 3 months.
Parents - Peter Quigley and Mary Reilly.
Interred in E931 on 30th March 1909.
HELEN QUIGLEY, Waterloo, aged 18 months.
Parents - Peter Quigley and Mary Reilly.
Interred in E931 on 2nd April 1910.
Also found this one and I'm guessing he must be from the same family -
MICHAEL REILLY, Waterloo, aged 18 months.
Parents - Joseph Reilly and Mary Ann Smith.
Interred in E3240 on 6th April 1914.
Plot owner listed as Joseph Reilly, coalminer, 14 Barntonhall Row, Waterloo, Wishaw.