I recently had to do something like this on a large scale.  Here's a simplified version of the script I was using to handle this.

//you can't create a new task record where the ownerid is inactive - grabbing a default user to set these to in this case
User default_user = [select id,name from user where name = 'Name of an active user here'];
Map<ID,User> inactive_user_map = new Map<ID,User> ([select id,name from user where isactive = false]);
//id of the record with activity history you want to copy
ID source_id = '00QV0000007xP5PMAU';
//id of the record you want to copy the activity history to
ID target_id = '00QV0000008O6wg';

List<Task> cloned_tasks = new List<Task>();
//you'll probably want to adjust the fields pulled in here to cover the custom fields and items you need in your instance
//whoid would cover a contact or a lead,  if you're targeting something else,  you'll want to use whatid
//ALL ROWS is necessary to get the closed tasks
List<Task> existing_tasks = [Select ActivityDate,CallDisposition,CallDurationInSeconds,CallObject,CallType,
  CompletedDateTime,CreatedById,CreatedDate,Description,Id,IsArchived,IsClosed,IsDeleted,IsHighPriority,IsRecurrence,
  IsReminderSet,LastModifiedById,LastModifiedDate,OwnerId,Priority,RecurrenceActivityId,RecurrenceDayOfMonth,
  RecurrenceDayOfWeekMask,RecurrenceEndDateOnly,RecurrenceInstance,RecurrenceInterval,RecurrenceMonthOfYear,
  RecurrenceRegeneratedType,RecurrenceStartDateOnly,RecurrenceTimeZoneSidKey,RecurrenceType,ReminderDateTime,
  Status,Subject,SystemModstamp,TaskSubtype,Type,WhatId,WhoId
  From Task t where whoid = :source_id ALL ROWS];

for(Task t : existing_tasks) {
  //use the flags that make sense for what you want to accomplish here
  task clone_task = t.clone(false,true,true,true);
  //once again - use whatid depending on the scenario
  clone_task.whoid = target_id;
  if(inactive_user_map.get(t.ownerid)!=null) {
    //Task has an inactive owner - switch to the default we decided on above
    clone_task.ownerid = default_user.id;
  }  
  cloned_tasks.add(clone_task);
}

System.debug('Coned Tasks : ' + cloned_tasks.size());
if(cloned_tasks.size()>0)
  insert cloned_tasks;