Intellij IDEA: Make refactor > Rename not select the whole word

It has changed name since the other answers, in 2021.2.3, It's still under Settings -> Editor -> Code Editing -> Refactorings however it's now named Preselect current symbol name for Rename refactoring.

You can do this by going to Settings -> General -> Refactoring and just deselect "Preselect old name" or just type Refactor in search box in settings so it will automatically go on above menu.

Aaaand it moved again!

It's probably the best to just search for Refactorings or Preselect old name, as it might move again in the future.

In my version 2020.1.2 it is under

Settings -> Editor -> Code Editing -> Refactoring

From IntelliJ 2018.2.6, the checkbox to disable is still "Preselect old name".
That is located in :

"Settings" menu
-> "Editor" section
   -> "General" subsection 
   -> "Refactorings" part

You can also just enter "refactorings" in the search box to find it.