Controlling 24-bit parallel RGB LCD with 16 bits
If you wanted to convert a 5 bit (0-31) to a 8 bit (0-255) number mathematically, what would you do to appropriately scale the range of value?

You would just use the rule of three: out = in × 255 / 31 = in × 8.22580645....

Now, look what your third solution does: it shifts the input by three bits left (because you insert three new bits at the right end): this means multiplying by 8. And, because the new bits you added are actually a copy of MSBs, it is equivalent to then adding the input, shifted to the right, by 2 bits: this means dividing by 4 (if we ignore rounding). See this for a better explanation:

           |R4|R3|R2|R1|R0|    input
------------------------------------------------------------
  |R4|R3|R2|R1|R0| 0| 0| 0|    input shifted left by 3 (x8)
+                |R4|R3|R2|    input shifted right by 2 (/4)
= |R4|R3|R2|R1|R0|R4|R3|R2|    output


So the equivalent expression is: out = in × 8 + in / 4. Wich is out = in × 8.25

See? This is almost the same thing, mathematically.

Basically, this method will allow you to take full advantage of your LCD contrast: full black and full whites are maintained (0 translates to 0, 31 translates to 255), and it tries to evenly distribute the steps because the scaling factor is constant (the 8.25 ratio). Of course because of rounding (R1|R0 have been left out during the shift right operation), some steps are a bit bigger than others, but you can't avoid that.

Method 1, in comparison, will not allow you to get full white, but all steps will be equal. In my opinion, rounding errors will be less noticeable than a greyish white.

I'm not sure I understand you method 2, but if I do, the steps will have more errors and be much more irregular than method 3, because it does not translate to a mathematical expression that is just a simple linear scaling of the input by a constant factor (and rounding). There is no advantage to that method whatsoever. 

In fact, anything other than method 1 and method 3 will have no advantage.