The best argument for diplomatic immunity is that it makes diplomacy in general possible.
Diplomacy is very important in our world today. With so much interaction between countries, we need to have diplomats in order to do our best to keep those interactions amicable. If we did not have diplomats, it would be much more difficult for countries to get along.
The tradition of diplomatic immunity helps make diplomacy possible. Countries might not send diplomats to other countries if they feel the other countries would harass the diplomats legally. This would be especially true for countries that are not exactly friendly with one another. Immunity makes it so that countries do not need to worry about this.
Thus, the best argument for diplomatic immunity is that it contributes to world peace by making diplomacy possible.