Hi 'heiserca'
I have no connection to your family but I just happened to notice your post and thought I'd attempt to help you out.
FIRST: I think your post could possibly be better located in the LANARKSHIRE section as that's where Lesmahagow is and you think that was where Eliza was born.
HOWEVER: ELIZA MORTON SYME married one CHARLES
CLARIE not CLAZIE in Inverkeithing Parish/District, Fife in 1866. The record is available on Scotlandseople and will give you her parents' names.
UNFORTUNATELY: There doesn't appear to be a Birth for Eliza in LESMAHAGOW, LANARKSHIRE although ScotlandsPeople does put up TWO matches in Lanarkshire between 1 Jan 1836 and 31 Dec 1838.
HOWEVER: There is an ELIZA MORTON SYME born in CERES, FIFESHIRE on 6 DECEMBER 1835 in the Old Parish Registers on ScotlandsPeople who has one parent named WALTER. The record is available.
ANCESTRY 1871 Census has the couple listed as CLAZIE (no surprise there

) living at 8 Portland Place, Leith North, Edinburgh. 1881 she's LIZIE CLAYCIE and 1891 she's ELIZA M CLAZIE born in CERES, FIFE and in 1901 she's ELIVIA CLAZIE again born in CERES, FIFE. As a Fifer, I can tell you that CERES and INVERKEITHING are at opposite ends of the County and a long, long way from Lesmahagow in Lanarkshire. There are Census results for Eliza from 1851-1901 and also a Public Tree on the Ancestry site. I think the 1851 entry is wrong though
I think you may need to re-think your results

Feel free to ask any further questions.
Regards
Chiad Fhear