Trying the simplest kernel module from LDD3 without any modification on custom built kernel v4.1.0-rc6 for Beagle Bone board with BusyBox v1.23.0. The code for the module is as follows:

#include <linux/init.h>
#include <linux/module.h>
MODULE_LICENSE("Dual BSD/GPL");
static int hello_init(void)
{
    printk(KERN_ALERT "Hello, world\n");
    return 0;
}
static void hello_exit(void)
{
    printk(KERN_ALERT "Goodbye, cruel world\n");
}
module_init(hello_init);
module_exit(hello_exit);

The Makefile is:

ARCH := arm
CROSS_COMPILE := arm-cortex_a8-linux-gnueabi-
obj-m := hello.o
all:
        make ARCH=$(ARCH) CROSS_COMPILE=$(CROSS_COMPILE) -C /path/to/linux/source/tree M=$(PWD) modules
clean:
        make ARCH=$(ARCH) CROSS_COMPILE=$(CROSS_COMPILE) -C /path/to/linux/source/tree M=$(PWD) clean

The module is compiling and installing on the rootfs just fine. also it is loading:

$ insmod hello.ko 
[   30.692404] Hello, world

But when trying to remove it, I am getting:

$ rmmod hello.ko 
Segmentation fault
$modprobe -r hello.ko 
Segmentation fault
$ lsmod
hello 813 0 - Live 0xbf000000 (O)

The kernel is compiled with module unloading (both regular and forced) support enabled.

What can be the possible cause of this issue? What is the way to approach the investigation of it?

Update:

As suggested in the comments I have tried including linux/kernel.h, defining the MODULE,LINUX and __KERNEL__ symbols. Added the__init and __exit prefixes to the functions. Removed the static modifiers. Removed the printk lines. The result the same. dmesg shows only the initial greeting. Loading and unloading of the kernel's modules such a gpio_keys or crypto/ccm is working, surprisingly. So the only thing remaining to suspect is the way the module is compiled..

Update 2
Updating the kernel to the freshest snapshot didn't help. Compiled the module with different optimization settings didn't help. Next step, I guess, I am going to modify the BusyBox's rmmod to have some indication of the problem location..