Hike up to the reservoir. Located on the hill above the whaling station is a fairly large lake. Footing may be muddy, but the scenery is nice and the view of the bay is excellent. Be aware that terns may begin attacking from the air; if they do it means that you are near their nest and should backtrack until they feel you are a safe distance away.
Visit Shackleton's grave. The cemetery is on the opposite side of the whaling station from the museum. Shackleton is buried here along with many of the whalers who died on South Georgia.
Visit Shackleton's cross. The cross is located at the end of King Edward Point, beyond the British Base. A trail to the cross is fairly easy to follow. While visiting be on the lookout for the very territorial fur seals; should one charge you it is best to carry a long stick or tripod with which to tickle its whiskers, which surprisingly deters most attacks.
Write letters home. There is a mailbox in front of the museum, and stamps and postcards can be purchased in the gift shop. Mail will reach most destinations within two or three weeks.
According to a recent program on BBC Radio 4 the ashes of Frank Wild, Shackleton's right hand man, have been placed alongside Shackleton's grave. Frank Wild was in charge of the men left on Elephant Island for months whilst Shackleton went for help.