Rooms are pretty much booked in Omaha at this time, by fans of teams hoping to get to Omaha. Day after the super regionals they are cancellations everywhere. Many folks book with the hopes they will be there and go for opening ceremonies(June 14). Also try right across the river in Iowa (Council Bluffs).
Try Best Western White House in Bellvue, just down the road from Rosenblatt. You also could ask the place where your son is staying to put you on a wait list. The further you go downtown the more bucks you pay.
Hope that this helps.