1 - Nearest Neighbor , dealing with same distance
The best approach would really depend on your application and what is important too you.

However, things you can try include:


Increase K until there is no tie anymore - if you increase to 2, you will likely have another tie, since they are at the same distance already. So, 3 or higher should do the trick.
Include another feature to your classifier - adding another dimension to your space could solve the problem if the values for both these data points are different.
Choose another distance metric - you could have a preferred way of measuring distance, but for ties, chose another metric that will break it.
Establish a rule for breaking ties, e.g.


Pick the class with the most observed data points
Randomly assign a class



More information on your application would help to choose a best approach.