Not sure if this is what you're looking for, but here goes...

If cell G5=Dog, H19=Cat, L4=Fish, AA42=Hamster...

Wherever your list is located, add a column before it, for this example I'll use column A to start, with my list in column B.

In column A, rows 1-4, enter the formula: =RAND()
In column B, enter formula references to the cells containing your data
in this case,
in B1 enter: =$G$5
in B2 enter: =$H$19
in B3 enter: =$L$4
in B4 enter: =$AA$42

(The $'s are so that when you sort, the cells are referenced statically, not relatively.)

Select columns A and B, then click the A->Z (sorting) button. Each time you click that button, the list will resort randomly. This is because the "RAND()" values in column A are volatile - meaning they change every time the sheet recalculates. With only 4 entries, the list can and will repeat the same order often (even with randomness). In a longer list it will appear much more random.

If that's not what you're looking for please try to explain in more detail.