Design 1

This is my first instinct design.  The obvious iCal and iStudiez inspired design.  This UI has a calendar that keeps track of all the chores' deadlines.  Along the left side is a color coded list of family members that matches color with the chores assigned them.  The right side has all the info on the selected chore, and allows the user to change it by filling out forms.  Advantages include high learnability due to external consistency and easy long term planning.  However the learnability can be a little over dependent on that consistency (no reason to believe this is a very learnable interface for a user inexperienced with similar apps) and the dependence on forms to change everything is likely inefficient.

Design 2

This design was an attempt to address some of the issues with the first design.  Specifically, I wanted to use a more Direct Manipulation approach and focus on short term planning and visibility.  The window across the top is a timeline.  Each block represents a chore.  The height shows its priority, The width is the current window it can be done in (the box starts at the planned time to do the chore and ends at the deadline).  The boxes can be directly manipulated to change chores, but selecting one also brings it up in the lower right and allows use of a menu.  In the lower left is a family menu, color coded and designed to show work distribution and presence or absence of conflicts at a glance.  

Design 3

This is my "stretch" design for illiterate users.  In the top left is a list of family members (by photograph) and icons that represent certain chores assigned to them (eg. laundry, cooking, shoveling snow).  The chore icons are located in the lower left and can be dragged over to family members to assign it to them.  When a chore is selected, its information is displayed in the right window.  This includes a calendar and a clock, to set the deadline, and a slider to pick the priority level.  I assume an illiterate user will have some experience with numbers, dates, and times; though I combine the visual display (clock, calendar) with a numeric display of date and time to make it as easy as possible.