My Lucario has a Lucarionite, but I still can't mega evolve and I have completed the game (This is in Ultra Moon).
Right - the most common problem is that you haven't received the Mega Ring from Sina and Dexio in Poni Plains. Use Charizard and Fly to the Poni Plains spot, then head into the cavern behind you. After a cutscenes and a battle, you receive the Mega Ring.
Otherwise your game is corrupted.
As I can't say definitely say without more information, here are all of the possibilities I thought of as to why you Lucario isn't Mega Evolving;
First off make sure you're Lucario is holding the Lucarionite as it can't mega evolve without it. Then when you start a battle with a wild Pokemon or a Trainer, select the fight option and on the bottom screen you'll see an option to Mega Evolve and once pressed and a move is chosen, Lucario will Mega Evolve and use said move that turn. As Rahat mentioned, you cannot Mega Evolve more than one Pokemon per battle, meaning if you Mega Evolve a different Pokemon before Lucario and then try Mega Evolving Lucario in the same battle, the game simply won't let you.
If you are interested in reading more on Mega Evolution, here is Bulbapedia's page on Mega Evolution, which helped me find/think of a couple of these answers that Rahat didn't previous mention in the comment below your question.
Hope I helped :)