I applied the first coat of polyurethane after staining and the polyurethane ran. I sanded the runs and it is slightly lighter in color. Should I reapply stain at this point?
Many finish coats, including polyurethanes, can slightly darken a finish when applied. Your sanding, if it did not cut into the stained surface, may have just removed that darkening poly layer.
You should try applying a coat just to the area you sanded, feathering the edges with a nearly dry brush. Let it dry and see if the color has evened out.
If not, I would recommend sanding the whole side. If it has lightened significantly, re-stain the side and then re-poly. If not, just re-poly.
You may want to look at some of the wipe-on finishes. These are satin finish (not high gloss) and go on easily with much less chance of runs.
You can try adding stain. Its hard, because PU (PolyUrethane) seals the wood surface. You might have luck with a gel stain or make a glaze overcoat, which will overcoat onto lighter area. Glaze would be my preferred technique, as you can 'play' with the effect, because of the extended drying time.
You would then overcoat with a new (lighter) coat of PU.
Tinted polys are another route (so called all-in-one stain & PU).