Throughout Monster Hunter Stories 2, you will be able to gather several eggs, all of which you can hatch into new monsties you can take into battle. You can also give these monsties names, but if you ever want to rename them, it's also possible. In short, if you want to rename monsties in Monster Hunter Stories 2, you will need to find a stable.
You can find a stable in the town area you first encounter in the game, as shown above. To use this stable, you will need to talk to the felyine called Stable Paw; doing so will open up a menu afterward. Once this menu pops up, you will then need to click the option that says manage monsties.
After clicking this tab, you will be able to rename your monsters by finding a monster you wish to rename and clicking on it from a list by hitting the button for manage. You find out what key this is for you by looking for the key at the bottom right of your screen.
When you click manage, you will be presented with two options - one of them will be to rename the monster. To rename the monster, click on the option that says rename and then choose a new name and click ok.
For more on Monster Hunter Stories 2, check out How long does it take to beat Monster Hunter Stories 2? and How to find The Flying Fury in the (Hard) Drifting Balloon Quest? | Monster Hunter Stories 2