The use of real names during an event is discouraged, and members are typically given a new "hash name," usually in deference to a particularly notorious escapade, a personality trait, or their physical appearance. In addition to being funny, this gives the hasher a certain degree of anonymity when the hash trash is posted online!
Newcomers referred to by their given name, typically with "Just" prepended to their first name. Members are named at significant milestone: a certain number of attendances, after they hare their first trail, or other noteworthy event. During a naming event, the individual is singled out and asked multiple and typically embarrassing questions. Other members may then share stories or observations about the individual, with the final name being chosen by general consensus from multiple suggestions.
Whatever you do, don't try to pick a hash name yourself. Your name is chosen for you. If the pack thinks you like a name, then they'll probably pick something else. Don't forget to bring your sense of humour!