I'm not sure I understand what you mean. You say a 'specific Matlab Figure' which implies you already know your figure, yet it is the figure you are naming. Line objects do not have a 'Name', only a 'DisplayName' and this would generally be different for multiple lines on the same axes as it appears on the legend, though I guess it can be the same for multiple lines.
So what exactly are the criteria you are trying to search for?
Obviously, as with almost anything of this nature, it is usually far easier to keep track of these things on the raw data before/when you do the plotting rather than a post-plotting slow search of graphics objects.
