How to make this random number game better?
A few observations:


In place of the while loop with you managing the numAttempts index,  why not use a for loop with the in keyword? i.e. for numAttempts in range(3):
Note that with the above, you can still use your else block (for supports the else clause! :)  )
Multiple games keep asking for your name. Unless your code has Alzheimer's, consider avoiding that :) 
Related to the above : Structurally, you've chosen to recurse to play another game. Why not keep your game atomic and keep the whole offer another game choice/logic external to your function? 
Your second elif does not make sense. Why not just else (guess can only be &lt; number now!)