python -- apply/run script for every array instance individually

Distribute copies.

Further to @moonboots answer can copy and distribute the object. Each copy shares the one mesh.

import bpy
from mathutils import Vector
from random import uniform


def array(context, count, rel_offset=Vector(), rnd_var_range=(0, 0), rnd_var_dir=Vector()):
    scene = context.scene
    for ob in context.selected_objects:
        mw = ob.matrix_world
        R = mw.to_quaternion().to_matrix()
        offset =  R * rel_offset.dot(ob.dimensions) * rel_offset.normalized()
        copies = count - 1
       
        for i in range(copies):
            copy = ob.copy()
            t = copy.matrix_world.translation
            t  += (i + 1) * offset            
            copy.delta_location = uniform(*rnd_var_range) * rnd_var_dir
            copy.select = True
            scene.objects.link(copy)

# test call

context = bpy.context

array(context, 
    count = 5,
    rel_offset = Vector((1.1 , 0, 0)),
    rnd_var_dir = Vector((0, 1, 0)),
    rnd_var_range = (-1, 1))
    
array(context, 
    count = 5,
    rel_offset = Vector((0, 0, 1.1)),
    rnd_var_dir = Vector((0, 1, 0)),
    rnd_var_range = (-1, 1))

Result of running script with Suzanne selected. The variation of both arrays in test call are in local Y direction.

The same script updated to Blender 2.8 API

import bpy
from mathutils import Vector
from random import uniform


def array(context, count, rel_offset=Vector(), rnd_var_range=(0, 0), rnd_var_dir=Vector()):
    scene = context.scene
    for ob in context.selected_objects:
        mw = ob.matrix_world
        R = mw.to_quaternion().to_matrix()
        offset =  R @ (rel_offset.dot(ob.dimensions) * rel_offset.normalized())
        copies = count - 1

        for i in range(copies):
            copy = ob.copy()
            t = copy.matrix_world.translation
            t  += (i + 1) * offset            
            copy.delta_location = uniform(*rnd_var_range) * rnd_var_dir
            context.collection.objects.link(copy)
            copy.select_set(state=True)
            context.view_layer.objects.active = copy
            
           
            

# test call

context = bpy.context

array(context, 
    count = 5,
    rel_offset = Vector((1.1 , 0, 0)),
    rnd_var_dir = Vector((0, 1, 0)),
    rnd_var_range = (-1, 1))

array(context, 
    count = 5,
    rel_offset = Vector((0, 0, 1.1)),
    rnd_var_dir = Vector((0, 1, 0)),
    rnd_var_range = (-1, 1))

Main changes are:

• @ operator is now used for matrix/vector component-wise multiplication
• selection of an object in context is now done with copy.select_set(state=True)
• The link must now be done via collections and not view_layer

set the object name to something meaningful like "bar" or whatever. apply the modifier ass it is with no Z variation. then press [TAB] to enter edit mode, press [A] to select all, press [P] and select separate by loose parts. press [tab] back to object mode. and run:

import bpy, random as r
objs = bpy.data.objects
for i in objs:    
    if "bar" in i.name:    #"bar" name of the array obj "Cube" to you n
        i.select=True       
        bpy.context.object.location[2] =1+r.randint(-20,20)/10 #"here your random funct"

...i think this should do the thing.

I am not sure if you can change the data for individual instance of the array modifier, since, I guess, the instances share the same data block.

One option is to apply the modifier, so as to combine the data of all objects into the parent object. This will complicate the calculations, though. The other option would be creating separate object for each instance. The most obvious way to do this is to copy the mesh data mesh = bpy.data.objects[x].data.copy() and use it to create a new objectobj = bpy.data.objects.new('new_obj', mesh) followed by bpy.context.scene.objects.link(obj). The objects location can be changed after it's linked into the scene.