Can lag bolts go fully through both pieces?
Are you replacing all of the treads?  If so, can you tear out the old, rotten stringer so you don't need to sister in a new one?  That would be best if the part of the stringer supporting some treads is beginning to fail.  It will also make it easier to use a modern connector between the top of the stringer and the joist supporting the landing.
If you can't tear out the old, I think framing nails are a better choice than lag bolts.  They'll distribute shearing forces across more nails and therefore more wood.
Lag bolts are often used to hold a beam together but it's the ends of the bolt and the outside faces of the beam doing the work.  In your case, you want the fasteners to transfer load from one compromised stringer to the new one; and you want to spread that out with several framing nails, instead of creating a point load inside of the stringers with fewer (but larger) bolts.
Be sure to use galvanized or stainless steel nails since it's an outdoor project.  If you're unsure which, your local building supplier will know (and they probably won't even sell stainless nails if you don't live near saltwater.)