I'm doing some research into that specific period in time but I was just wondering if there was like a general start point and end point to this period before one could say that Norman influence was no longer quite as powerful in that region.
Although there were Normans fighting for the Lombard Dukes of Benevento and Salerno for about a century prior to this date, I would put the start date at Robert Guiscard's conquest of Sicily in 1061.
Following Tancred's usurpation, Henry VI pressed his wife's claim (Constance of Hauteville) after Tancred's death at 55, when his nine-year-old son William III was left on the throne. When Henry VI was crowned King of Sicily on Christmas day in 1193 after having taken Palermo earlier that month, I think the period of Normans ruling Sicily can be said to have come to an end, although their influence would be felt for long after.