C++ Adding Two 16 bit binary numbers together
A few issues here:


The way you “rename” the arrays is wrong, and furthermore there is no
point in doing such renaming.
There is no reason to divide by 10 (ten, in decimal) within your
code: as everything you do is binary, the number ten has nothing to
do with what are trying to accomplish.
While doing the addition, you forgot to loop over the bits.
You shouldn't return 0 in a function of type void. In fact, you
shouldn't return any value.


Fixing those issues should give you something similar to this:

void Solution() {
    lcd.clear();
    lcd.setCursor(0, 0);
    lcd.print("Solution");
    lcd.setCursor(0, 2);

    // Compute the sum.
    int sum[16];
    int remainder = 0;
    for (int i = 0; i &lt; 16; i++) {
        int s = binaryOne[i] + binaryTwo[i] + remainder;
        sum[i] = s % 2;
        remainder = s / 2;
    }

    // Print it.
    for (int i = 15; i &gt;= 0; i--) {
        lcd.print("%d", sum[i]);
    }
}


Note that there is no point in computing a 32-bit result if you can only
print 16 bits: just accept that the computation is done modulo
216, just like a 16-bit ALU would do, or the unsigned int
data type in the Arduino.

Of course, the whole idea of doing the computation one bit at a time is
highly inefficient. See Majenko's answer for a more idiomatic and
efficient way of managing numbers bit-wise in C and C++.