Adding together what you have told us and what I have found by looking at the 1881 census, she was born in the city of Edinburgh in 1840/1841, and she was sufficiently literate to write her own name. She is very consistent in giving her age in the census - 20 in 1861, 30 in 1871, 40 in 1881.
If, in spite of of being able to write and confident about her age, she did not know the names of her parents, she must at least have been born illegitimate, and possibly even abandoned as you wondered.
The City of Edinburgh parochial board would undoubtedly have come across her in these circumstances, and would have collected a good deal of information about her. Unfortunately the Edinburgh parochial board registers of applications and registers of poor have not survived. There is a possibility that their minutes, which have survived, might contain a clue, but as the parochial boards were not set up until 1845, there may not be much reference to her. The surviving parochial board records are in Edinburgh city archives.
I think Lauriston is in the parish of St Cuthbert's. If so, and if Eliza came to the notice of the kirk session, and if the kirk session took an interest in her, and if they recorded information about her in their minutes, and if that record has survived, it is likely to be available in digital form in the historical search room of the National Records of Scotland in Edinburgh. But that is a lot of ifs.
