AVR Freaks

You're biggest problem is that you are defining and initializing variables in a header file and then including your header file in multiple c files.  Also, your definitions are outside of your header guard.  That's not causing any build errors, but it's a bad habit to get in to.

Your header file should look more like this:

#ifndef LCD1602_I2C_H_
#define LCD1602_I2C_H_

extern uint8_t arr[] , arr_size;

// the rest of your code goes here

#endif

The actual declarations and initialization of those extern variables should be in your LCD1602_I2C.c file something like this:

uint8_t i, bitmask = 0x08;

uint8_t arr[]={0x48, 0x65, 0x6c, 0x6c, 0x6f, 0x20, 0x41, 0x72,
	0x64, 0x75, 0x69, 0x6e, 0x6f, 0x20, 0x3a, 0x29, 0x20, 0x57,
	0x68, 0x61, 0x74, 0x27, 0x73, 0x20, 0x75, 0x70, 0x21, 0x20,
	0x48, 0x6f, 0x77, 0x27, 0x73, 0x20, 0x63, 0x6f, 0x64, 0x69,
	0x6e, 0x67, 0x20, 0x74, 0x6f, 0x64, 0x61, 0x79, 0x20, 0x3a,
0x29}, arr_size=sizeof(arr);

Lastly, you are missing a return statement in your BF function.

Hope that helps.