But why would they have persecuted her without the name? If her name wasnt Ultimecia, say she was just an ordinary child who eventually becomes defiant and takes the name Ultimecia, why would SeeD have a reason to persecute her? She wouldnt have even been a sorceress at birth, yeah?

It's not like SeeD has a lot of information on Ulti, so they wouldnt be able to predict her arrival... by that I mean, they dont know to who she was born, or where, or win. So why would she be persecuted by SeeD from birth unless her name was Ultimecia... and then we arrive at the point of why someone would be named Ultimecia, and whether or not the legend of Ultimecia was known.

I'd say more likely she was persecuted when she became a sorceress (this is assuming that her name was not Ultimecia from birth), and then took the name Ultimecia as this challenge of fate.