The island itself is genuinely photogenic (Gul Mahal, the elephant statues) but there's not much to actually 'do' once there beyond the restaurant - worth it for photographers and honeymooners, skippable if you've already done the Lake Pichola boat ride, since most Pichola boats include a short stop at Jag Mandir. The Taj Lake Palace is not a sightseeing stop - access is normally for hotel guests and confirmed reservations only.
Construction began in 1551 under Maharana Amar Singh and was completed decades later by Maharana Jagat Singh I, after whom the palace is named. Its first structure, the Gul Mahal, sheltered the Mughal prince Khurram during his 1620s rebellion against his father Jahangir; Khurram later became Emperor Shah Jahan and is said to have drawn architectural inspiration from Jag Mandir for the Taj Mahal. The island palace again served as a refuge in 1857, sheltering European women and children during the uprising against colonial rule.