Does anyone have any good tips for cleaning and brightening the engraved steel surfaces on a shotgun?

I'm not sure what you mean. If you mean protecting them from rust, simply wipe them with a decent gun oil, or similar, just be sure you use a soft cloth to get it down into the depressions if you've handled it with sweaty hands.

AFA "brightening" goes, I would suggest you leave it alone, depending on what is causing it to darken. If it is just overall darkened for some reason, you might try a fine soft toothbrush with Flitz or JB. Without seeing it and not really knowing what your asking that's about as far as I will go. My first response is to leave it alone unless there is a specific reason to be more proactive.

If it is rusty, you might VERY CAREFULLY use some fine steel wool on a tooth pick or similar to work on the rust spots, followed by wiping clean and then oiling.

Thanks for the input. I am not contending with rust, and do wipe regularly with a very light coating of gun oil. It's just that the steel around the engraved areas looks dull --- I was concerned that the wiping with oil was causing the dinginess. I didn't want to consider fine steel wool until I got some input as to whether that would mar the underlying metal.

It could be an oil buildup/film. Try using some WD40 as a cleaning agent, that is maybe put some on the area and let it sit for a while and then wipe it off, and see if that cuts it. AFA steel wool goes, I wouldn't use it unless absolutely necessary. You will almost certainly change the appearance of the metal if you rub much and it won't match the areas around it.