The button can be greyed out if there are simply no plots to plot or you're playing an underage character. You should still be able to kill almost anyone if you go to their personal diplomacy screen and press the small circle with the Intrigue logo on it, but it won't work against your direct children as far as I know. Also the bigger you get, the more plots you will get. Generally they'll just be "Plot to Revoke Title X" or "Kill Baron Y", generally the "Kill Person" plots benefit you in some way, even if it's not apparent to the player immediately.