what is key or critical to mat?

ta'veren-ness
the horn
moraine's rescue
the seanchan (via tuon)
gambling
the band
insane amounts of battle knowledge

verin knew he is ta'veren, she may have deduced that he blew the horn/that it is in the tower, she would have no idea about moraine's rescue (unless lanfear happened to mention it in passing, which i doubt. but of course mat, moraine and verin are connected together), she may have deduced some seanchan connection (though that is doubtful), she may know about his luck, she may know of dark friends in the band and she may have deduced his battle/fighting acumen.

but....he is on his way to caemlyn, so it is entirely possible that verin is using mat to accomplish things on her behalf. or she knows of a plot on rand and knows that mat will help to save him.