Making a game is all about weighing pros and cons. If you prefer, you can think about it in terms of cost benefit analysis. This starts with picking a game engine.
While there are the well known like Unity or Unreal Engine, there is the less known like Construct 2 or Coco 2Dx, and there is also the option to create your own. To be perfectly honest, you could also go without an engine and just create everything that you need as you need it. There are pluses and minuses for each of these choices so how do you really decide what is best?
The first thing to look at is why you are making the game. If you are making the game in order to showcase your skills to future employers, then it would be best to use one of the more popular choice. You want to show the world what you can do with the tools that they are most likely to be using. There is no point in showing them a game made with Coco 2Dx, if they are only going to be making 3d games using Unreal Engine. Since there are so many engines out there, it is best to make a variety of games with different engines, if this is truly your goal.
If you are making games in order to experiment with technology, then it is best to use the lesser known engines. There is enough information out there that would allow almost anyone to make a basic game in Unity. Try using Löve. What about Moai? While these might not say much to most people, those in the know might still be impressed. These more obscure game engines also give you an excuse to learn a new programming language!
Finally, you can create your own engine, or go without one predefined. This is the hardest route to take. You are forgoing the knowledge and time that others have invested in order to reinvent the wheel on your own. If you can do this, it shows that you are capable of truly great things. Programming a game engine from the ground up requires foresight, planning, and immense testing. There is the documentation, the maintenance and feature requests from others, and the tutorials on how to use it. This can easily be a project bigger than most games.
As you can see, choices abound when you start looking at it. While the basics are good, you have to think about what you are trying to prove with your game. Who are you trying to impress?