What should I cover gap with below stair banister

Since that area of the floor will be under the new cable banister/rail, there would be no fear of creating a tripping hazard by covering the gap with a decorative wood strip. I would use a low-profile oak "carpet transition bar" type of strip, stained and varnished to match.

The simplest way to fix the issue is what Jimmy Fix it suggests, or even a smaller version of what he has pictured by cutting down that material even smaller before finishing the same as the floor.

If you want it to look the way it does now minus the gap, that will be an entirely different matter.

It will require the removal of the original nosing, cutting the faulty flooring that met the old nosing with a router and straight edge to create a straight line cut to meet the new stained-to-match nosing. The new nosing will be a ripped down 5 1/4" nosing to the needed width of the old 3 1/2" nosing plus the cut needed to correct the edge, say maybe the 5 1/4" nosing would be recut to 4" to fit the newly adjusted space.

The explanation I just gave is truly an over simplification of what needs to done. If you are up to it and can handle it, I will elaborate in an edited answer. If it cannot be handled, let me know, that way the short novel I would have to write to guide you through it will not be needed. That way I will or would direct you to hire a faux finisher that can fill the gaps and fake in the wood grain to make the gap go away. I have seen it done before with a staircase that had wood balusters with 3/4" drill holes that needed to go away, since there were no longer going to be there. Not all the wood plugs matched the grain to blend in well enough in all 40 or 50 holes that were filled so a faux wood finisher was hired to wood grain the remaining ones so they would blend in with the surrounding areas.