My answer is going to avoid drone technology because I don't really know anything about it. I'm just going to assume it's like throwing a ball straight up in the air, but somehow the ball reaches it's peak and stays at that horizontal level.

It depends on what you mean by 'remain where it is'. There are two reference frames that are important here, and we will place two observers in these frames. The first observer (Observer A) is stood on the platform in the train station. The second observer (Observer B) is sat on a chair on the train.

Consider first the case where the train is stationary with respect to the outside world, and is stopped in the train station at the platform. Observer B is holding the drone, and Observer A looks in. Right now, everything is in the exact same (inertial) reference frame. Observer B puts the drone on hover mode, and it will be stationary with respect to both observers.

Now, the train accelerates forwards (with respect to Observer A), moving away from the station. Observer B would see the drone accelerate backwards. Observer A would see the drone as stationary, but of course would see the train accelerate forwards. I hope this answers your question.

To further this answer, we can also think about the case where the train is moving but not accelerating. Lets say now Observer A is still stood at the platform, but this time the train is passing through at constant speed v. If, a little before the train passes through the platform, Observer B was to put the drone on hover mode (whilst the train is moving at speed v and not accelerating), the drone will continue to be stationary in Observer B's frame of reference.

As the train passes the platform, Observer A will look through the window of the train and see the drone moving forwards with speed v (so it is moving along with the train).

Long story short, as long as there is no acceleration, the drone will stay stationary with respect to the train (Observer B). It is only as the train accelerates forward (backward) with respect to the direction of motion of the train that the drone will appear to accelerate backward (forward) with respect to the observer on board that train.