My exterior stairs have three stringers about 16 feet long each. A couple of sections are rotted.
Is there a way to splice a repair, or must the stringer be a continuous piece of lumber?

Those are long stairs! There would be a couple of ways to do a splice: nail some stringers into a beam and/or support the joint with a post.

With rot problems, wouldn't it be easier to rebuild the whole staircase?

I would not try to repair a rotting stair. The rot will continue if you splice in a new piece for support. I'd bite the bullet and build new. If you repaied that stair and someone were to get injured your home owners insurance is not required to cover your losses.

Build New !

I agree with the above and replace all of it, for peace of mind over all