As stated in the comments, you should use the response linked in the comments, which is higher than mine and does not require editing the original JavaScript.
If you open fullCalendar.js and look at the very first sections of the file, there is something marked as
there you will find something like
buttonText: { prev: ' ◄ ', next: ' ► ', prevYear: ' << ', nextYear: ' >> ', today: 'today', month: 'month', week: 'week', day: 'day' },
Change today to
today: 'LOL!',
et viola!
If you want to dynamically change text on the client side, use jquery like this
$('.fc-button-today span span').html('lollllll');
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