Ok. Nameless Fraser boy is a no; his mother was Robina Fraser, and there was a correction added in Feb 1865 that confirmed his father as Alexander Chapman.
1851 census - I think it's "Inverness-Shire Petty" (part of Scotland'sPeople original attached. I would've added the whole page, but there are file size restrictions. Just in case: Crown Copyright, National Records of Scotland).
Got the Cromarty prison records - Eliza Fraser was locked up for 10 days ending 1 Oct 1861, so she wasn't in jail for the census. Arrested for "stealing cloths"; she originated from Cromarty, Cromarty.
For the 1861 census I wondered if they'd misread her age, so I looked at all 13-19 year old female Fraser records for Ross & Cromarty, lol, but no luck placing Eliza. I've now started on Inverness (sounding more likely if her parents married there). There are 12 "Eliza Fraser's" listed (age 11-19) and four are aged 17.
So I have a few things to look up in Inverness apparently - much better than a dead end