Nothing to do with legal or illegal drug use!
No, the idea is to have five or six people (the hares) set a trail for others (the hounds) to follow.
The idea originated with expats working in Kuala Lumpur who, once they had run the trail, resorted to a local bar called the Hash House for refreshment. Hence the name 'hashing' was born and expats working in other countries such as Saudia Arabia and many other countries formed similar running groups. In Riyadh in the mid 80s I ran with the Suleimania Hash House Harriers (HHH).
Suleimania Hash awarded tankards to runners for completion of every 25 runs, but runners also had to act as hares and set a number of trails as well to get them.
Find out more by clicking here to read an article on HHH in Wikipedia, and here to read the guidance on setting a trail in the Saudi Arabian desert around Riyadh at that time (It's worth it just for the artwork!)