(My apologies if I am using the terminology incorrectly.)
Your terminology is correct, no problem.
Since I cannot just replace the bottom rail as there is no good bottom mortise joint to work with, my thought was to cut off the bottom stile mortises and just add a new board to the bottom of the stiles with dowels, glue or maybe pocket holes. Would this be too weak a joint? The bottom rail is 1-1/4"x5" and the stiles are 3.5"x1-1/4". My concern is that since the bottom rail would no longer be "sandwiched", that is would not be strong enough? My only other option seems to be to rebuild the entire screen(s). Does anyone have any suggestions? Thanks
Yes, you COULD do it that way, and yes, it would lack strength. But it may work, since the screens are not necessarily subject to torque, movement, or tension. I'm assuming they are mounted, then left in place.
Essentially, you want to cut some length off of the stiles and replace the lost length with new wood.
Several other options: 1) the illustration which cabinetman submitted is called a slip joint. It is one variation of a mortise & tenon joint. You could make a similar joint, except with the boards end to end as needed in your case. I do this whenever I need a long narrow rail in a length I cannot purchase. Or when I do a bent laminate curve for a front edge of a shelf or counter, and I need to extend the front edge along the straight section of the shelf. I match the grain on the face and join them end to end with a slip joint.(or perhaps it has a different name when used end to end?) 2)You could make a half-lap joint, which could be glued and clamped tightly. That would provide a pretty strong joint. 3)If you wanted, you could get one of those router bits that cuts finger joints and use that to join them with.
So I think one question you might ask yourself is, how strong does it NEED to be to do its job?