For example, a halfling has advantage on an attack and rolls an 8 and a 1. The 8 misses. Does he get to re-roll the 1 using Luck?
Yes, the halfling gets to reroll the 1. (The halfling gets to reroll the 1 even if the 8 didn't miss.)
How abilities that allow rerolls — like the hafling Lucky trait — apply to advantage and disadvantage is not immediately obvious, so it's actually covered explicitly in the rules for advantage and disadvantage, even using Lucky as the example (PHB, p. 173; Player's Basic Rules, p. 57):
When you have advantage or disadvantage and something in the game, such as the halfling’s Lucky trait, lets you reroll the d20, you can reroll only one of the dice. You choose which one. For example, if a halfling has advantage on an ability check and rolls a 1 and a 13, the halfling could use the Lucky trait to reroll the 1.
So yes, you get to reroll the 1. If both dice came up 1 though, you'd only reroll one of them though, not both.