My MP3 player has a lithium ion battery - is it better to let it drain before charging or can I ....

You will get the best battery lifetime if you try to strike a balance between not charging too often, and not discharging too deeply. Generally, that means using it normally until the battery is around 30-40% discharged (60-70% remaining), and then fully charging it then.

Lithium Ion battery lifetime depends mostly on three things:

• How many charge cycles, and how deep the battery was discharged. Most Lithium Ion batteries have between 400-800 charge cycles before they start getting much weaker. So you want to minimize the number of charge cycles. However, it is less damaging to the battery to do a small charge cycle, so you will get more charge cycles if you don't discharge it too deeply.
  
• How old the battery is. Lithium Ion batteries age quite quickly, and no matter how carefully you treat them they generally only last 3 years or so.

• NEVER let a lithium ion battery go completely dead, and then let them sit around uncharged. That can cause an internal short in the battery that can cause a fire. To prevent, this lithium ion batteries have an internal circuit that permanently disables them if the charge gets too low, and the battery will not be usable after that. (When your battery meter show empty, there is really still a little bit of charge in it by design to protect it from happening, but if that little bit of charge is lost, the battery is permanently dead). This issue is why you can't go to the store and buy lithium-ion batteries in standard sizes like AA, because they require this special protection circuit.
  

Yes, let it drain first, so the charging process will be better.

you know it really doesn't make that much of a differnce

Draining it is pointless. Lithium Ion batteries have no "memory loss" like older battery types. It doesn't matter when you charge them.