What is the best way to pack cookies for mailing?

Use a long round tin that you can stack the cookies in, and cushion the top or bottom with parchment paper

In an airtight container lined with thick paper towel.   Then put that in a mailing box with some sort of padding such as foam wrap, bubble wrap or even an old bath towel. 

Honestly, my best advice is to ship them in a USPS Priority Mail box as they are very sturdy and arrive quickly.    And don't ship crumbly cookies or cookies that have icing on them.   Choose a recipe that isn't going to be brittle or melt.

I would put them in a sandwich bag, wrap in layers of bubble wrap, place inside a tin, more bubble wrap and then a box.