I've forgotten the formal proof for this, but hopefully this is correct:
Consider a D6 (for the sake of concrete language).
When you roll a 1, you reroll the die and keep the result. This produces an average value of 3.5, and happens 1/6 of the time.
When you roll a 2, you reroll the die and keep the result (even if it's lower). This produces an average value of 3.5, and happens 1/6 of the time.
When you roll a 3, you keep the result. This produces an average value of 3, and happens 1/6 of the time.
And so on.
This gives the following formula for the average of the D6: \$ (3.5 + 3.5 + 3 + 4 + 5 + 6) / 6 = 4.1\bar{6}\$.
Working similar formulas for the other dice, we get this table:
\begin{array}{lccc}
\hline
\text{Die} & \text{(standard) Avg.} & \text{GWF Avg.} & \Delta \\
\hline
\text{d4} & 2.5 & 3.00 & 0.50 \\
\text{d6} & 3.5 & 4.1\bar{6} & 0.6\bar{6} \\
\text{d8} & 4.5 & 5.25 & 0.75 \\
\text{d10} & 5.5 & 6.30 & 0.80 \\
\text{d12} & 6.5 & 7.3\bar{3} & 0.8\bar{3} \\
\hline
\end{array}
Dice are independent. 2D6 will have an average value of \$2 \cdot 4.1\bar{6} = 8.3\bar{3}\$.
Common weapon average damage (Great Weapon Fighting):
\begin{array}{lcc}
\hline
\text{Weapon} & \text{Avg. GWF dmg} & \text{improvement w/ GWF}\\ \hline
\text{Greatsword (2d6)} & 8.3\bar{3} & 1.3\bar{3} \\
\text{Greataxe (1d12)} & 7.3\bar{3} & 0.8\bar{3} \\
\text{Longsword (1d10)} & 6.30 & 0.80 \\
\text{Double-bladed Scimitar (2d4)} & 6 & 1 \\
\text{Smite (level 1, 2d8)} & 10.50 & 1.50 \\
\qquad \text{(+ weapon damage)} \\ \hline
\end{array}
Observations:
The ability works out to about a +1 to damage.
It scales to almost a +3 when smiting. The more dice you add (high level smite, for example), the better the ability.See errata, below
The bonus is "swingy." It can range from a -2 to a +10 on 2D6, for example.
Errata
In April of 2016, Jeremy Crawford ruled that additional dice from abilities like smite can not be re-rolled by Great Weapon Fighting.
Yes
"If the attack involves other damage dice" sounds like a pretty solid indicator that "Superiority Dice" would be multiplied, if a maneuver were used that adds the result as damage.
While not applicable to everybody, that sounds like an awesome way to add extra damage to multiply (not that superiority dice are really game-breaking for additional damage) after learning that an attack hit... but I'd allow it, regardless.
Best Answer
By Rules as Written: Yes.
Player's Handbook, page 72:
This is fairly explicit. If it's a damage die, and it's rolled for an attack you make with a melee weapon in two hands, then you get the re-roll. This works for Combat Superiority, Smite, bonus dice from spell buffs, and even Sneak Attack if you could figure out a way to sneak attack with a two-handed weapon (currently, there is no way to do so).
By Rules as Intended: Probably not.
Mike Mearls, one of the lead designers of DnD 5e, replied to a similar question on twitter about whether Great Weapon Fighting allowed you to re-roll smite damage. His reply was that he would rule that Great Weapon Fighting only applies to weapon damage, not smite or, by extension, combat superiority dice.
However, Mearls has stated before that he is by no means the final word on what is or is not "correct". His statement is specifically the ruling he would make, not an official response from Wizards of the Coast as a whole, so it should be taken with a grain of salt.
What Effect Would It Have
Ultimately, you have to decide for yourself whether you want to allow this to happen or not. To help aid that decision, I thought I'd share with you the math on exactly what effect this ruling would have, in either direction. For these calculations, I'm using a standardized Fighter build, and calculating how much damage, on average, they would deal to an average foe at any given level, including hit chance, critical hits, and any other relevant modifiers.
Without Using Combat Superiority:
With Combat Superiority and no reroll:
Combat Superiority with reroll:
As you can see, overall the difference is fairly minor. All told, the damage increase is never more than 5% of the total damage being done, and this is assuming that you're using Combat Superiority on all of your attacks, which is probably not what you want to be doing when you have 3-4 attacks each turn.
It's certainly a buff, but nothing to write home about, and if it makes your players feel stronger or more clever to be using their abilities in this way, then I'd say let them have it. The difference is small enough that, overall, it's unlikely to have any real impact on the game.