No! The shield is not a weapon, this is designed to be a sword and board feature.
Shield is definitely not a weapon (no weapon stats), and there are currently no enchantments to make it a weapon. This is a feature designed for a brawler style (hand free for grappling) or a sword and board fighter.
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.
Best Answer
The Great Weapon Fighting fighting style states that:
A lance has the Reach and Special properties (as per the weapons table on page 149 of the PHB) but does not have the Two-handed or Versatile properties so it doesn't not qualify for use with Great Weapon Fighting.