If you want to rename a file in OS X, you can select it and then do one of several things. The first is to press the Return key and the name will be highlighted for editing, or you can get information on the file and rename it using the Info window. A commonly used alternative to these is to click the name of the highlighted file with the mouse and after a small delay the file's name will be highlighted for editing.
While I have found using the Return key is the fastest way to switch between selected and renaming mode for an item in the Finder, there are times when using the mouse to enter rename mode can be convenient as well. Unfortunately some people may find when clicking the name that it takes perhaps a few seconds longer than desired for the renaming mode to activate.
This issue can be fixed by adjusting the double-click settings for the mouse (or trackpad). The time it takes from clicking the name of an item to when the name gets highlighted is directly proportional to the double-click delay in OS X. If the delay is set to be long, then it will take longer from when the file is clicked for it to go into rename mode. If you set the double-click speed to be short, then the delay will be shorter.
As a frame of reference, if you set the double-click speed to be the fastest, then when you click a selected file to rename it the renaming mode will activate almost instantaneously. However, if set at its slowest, then it will take around 5 seconds for the rename mode to activate.
In addition to long delays, a slow double-click setting may also lead to inadvertent opening of a file when you are trying to rename it using the mouse. My recommendation when setting the double-click speed is to set it to the fastest setting (the one with the shortest delay) and then use your system as you normally would. If you are finding it difficult to open files when double-clicking, then lower the setting one notch and try using the system again. Keep doing this until opening and other double-click behaviors work as expected.