I have a brick and mortar fireplace in my basement. It will probably never be used as a fireplace ever again. I'd like to hang something over the mantle. A not-very-heavy something. If it were a wall, I'd just stick a nail or a picture hanger in it. Aesthetically, it doesn't matter if the mount is in a brick or a line of mortar. I'm not really concerned about ever having to remove the mount, but I don't want to do anything that will compromise the structural integrity of the flue only because we get earthquakes and I don't want it to become unsound.

What's the best way to do this?

Drill a hole in the mortar with a masonry bit. It helps if you dip the bit in water from time to time.

Insert a plastic drywall anchor and then the screw that comes with he anchor.

Hang the item from the screw.

They also make metal brick hangers - I’ve used them for wreaths Outside - basically they wedge in between the brick spaces - I’ll see if I can find a picture.

I've used Steve's method.