Does an action to stand up from being knocked prone generate an attack of opportunity (like it does in some earlier editions), or is it only if the target leaves the attacker's reach?
The Player's Handbook only says that opportunity attacks are triggered when hostile creatures move out of your reach. (p. 195)
You can make an opportunity attack when a hostile creature that you can see moves out of your reach.
So, the prone target can stand up and even move, as long you don't go outside enemy reach.
This new rule also applies for ranged attacks, casting spells, etc. (PHB p. 195)
When you make a ranged attack with a weapon, a spell, or some other means, you have disadvantage on the attack roll if you are within 5 feet of a hostile creature who can see you and who isn't incapacitated.
In older editions, these actions would cause an opportunity attack, but not in D&D 5e.