One very specific situation I've noticed:
The game sometimes gets really confused about doorways.
I've had it happen with Stinger and Restorer Hive variants, I've certainly seen it with Blinder and Demolisher Firefly, and it happens to me with Striker as well. If you deploy on one side of a wide-open doorway, and the enemies are on the other, the skill won't "see" them. Same with healing from the Restorer.
Some specific places you can "test" it and see the frustration for yourself:
Capitol Hill Legendary - the chamber after you come inside the building from the lawn/plaza, clear the Statuary, then traverse the hallways, where Rogue Agents sometimes spawn. If you stand outside the door, even if you target an enemy successfully, your drone won't go through the doorway. Your agent has to be physically inside the room for your Drone to target.
Capitol Hill Legendary 2 - many of the doorways in the halls leading up to the chamber mentioned above won't let a Firefly through unless you physically cross that barrier and deploy it then.
Roosevelt Island Legendary - The section where, in the non-Legendary version, you meet Angela Woods for the first time, go between the two guard towers, and have to destroy the first fuel tank to proceed. That doorway that has the fuel tank between the two chain link fences will not let a Firefly through for anything. When you throw it, it'll just self-destruct once it tries to cross that doorway. There are a few other doorways in Roosevelt that behave the same way.
Iron Horse, Fieser - We run the "broken control room" strat. I can throw grenades at Fieser from the control room, shoot him, whatever...but the invisible barrier between me and him, even with the window broken, won't allow a targeting skill to target.
If your Drone won't target someone, check and see if there's a doorway between you and your enemy. If it's safe for you to cross it, cross it and try re-targeting then.
Other situations I'm aware of - Rain, fog, and sandstorms all greatly limit the targeting range of Drone and Turret. It's not that they can't target, it's that they won't target unless the enemy is much closer than usual. All you can do is wait it out and do your best until it clears up.
And yes, every now and then, it just plain sits there. Like many have said, you can destroy/redeploy. With Drone you can call it back to you by double-tapping the deploy button. Sometimes that kind of resets it for me to send back out. Frustrating when it happens, but at least the cooldown is fairly low if you have to pop it and start over.