Well, I have an answer -- and a new mystery!
The sound happened again when I put my car in reverse this morning preparing to leave the house, but this time I was listening more carefully and realized the noise was coming from the driver's door. Putting the car in reverse causes it to automatically lock the doors, and that's where the sound was coming from. I tested locking and unlocking the doors with the button on the driver's door and it happened again. So... Yay! Not a transmission issue, just the lock. But why did it start doing this now?
And here we have a new mystery. I looked at the area around the lock on the inside of the door, where it meets the frame -- not, frankly, a part of the car I think about or look at much -- and see a bunch of small dents in the metal part of the door where it meets the door frame. Hopefully my first-ever attempt to post a picture on this site will work.
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I have no clue when or how these dents occurred. There's nothing on the outside, which seems to rule out a break-in attempt. I have no recollection of ever having anything caught in the door that would make dents like this, and there are several of them (a couple of the smaller ones are hard to see in the photo). A couple times I have had a jacket sleeve get caught in the door when I closed it, but not lately -- I live in Hawaii, where jackets are rarely a thing -- and a jacket that's only cloth, no metal parts. I suppose the metal parts of the seat belt could do it, but I sure don't remember anything like that happening, and since the belt automatically retracts it's hard to see how it would happen. In any case, nothing remotely like that happened in the last couple days, which is when the noise started.