When you say that you "don't have ___ on me", it means that you don't have it right then:
A: Can I borrow a pen?
B: Sorry, I don't have one on me.
This phrase is useful because just saying "I don't have ___" can have two different meanings:
- you don't have it now
- you don't own it at all
You use "I don't have one on me" when people ask you for things like:
- business cards
- a pen
You can also say "I don't have one with me", but "on me" is usually better if the object is small and can fit in your pocket or purse.