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 = 4.1̅6
Working similar formulas for the other dice, we get this table:
Die Avg. Δ
--- ---- ----
d4 3.00 0.50
d6 4.1̅6 0.6̅6
d8 5.25 0.75
d10 6.30 0.80
d12 7.3̅3 0.8̅3
Dice are independent. 2D6 will have an average value of 2 * 4.1̅6, or 8.3̅3.
Common weapon average damage (Great Weapon Fighting):
Greatsword (2d6): 8.3̅3 (Δ1.3̅3)
Greataxe (1d12): 7.3̅3 (Δ0.8̅3)
Longsword (1d10): 6.30 (Δ0.80)
Smite (level 1, 2d8): 10.50 (+ weapon damage) (Δ1.5)
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.
The bonus is "swingy." It can range from a -2 to a +10 on 2D6, for example.