The easiest way to gain advantage on a death saving throw is to use inspiration.
If
you
have
inspiration,
you
can
expend
it
when
you
make
an
attack
roll,
saving
throw,
or
ability
check.
Spending
your
inspiration
gives
you
advantage
on
that
roll.
Alternatively, you could be a Wild Magic Sorcerer, and use Tides of Chaos. Well, probably.
Starting at 1st level, you can manipulate the forces of
chance and chaos to gain advantage on one attack roll,
ability check, or saving throw.
The 3rd- level spell Beacon of Hope explicitly grants advantage on death saving throws for its duration.
Fighters can use Indomitable to reroll a failed saving throw - not precisely the same as advantage, but pretty close.
Beginning at 9th level, you can reroll a saving throw that
you fail.
There are almost certainly more options that I've missed - getting advantage on things is a pretty fundamental part of 5e.
I think this is pretty cut and dried with just what you have quoted. If you're a Luck Halfling and you roll a 1, you can re-roll it once and use the number. Then if you just don't like any number you rolled, even the re-roll, you can spend a luck point to re-roll (again). One doesn't replace the other, you simply have both features, and both operate exactly as they written.
First Case: Feature then Feat
So if you use Halfling Luck to reroll a 1, can you then decide to use a luck point and still get to choose any of the dice?
Yes.
Example
Let's say you roll to attack, and roll a natural 1. Your halfling luck kicks in:
When you roll a 1 on an attack roll, ability check, or saving throw, you can reroll the die and must use the new roll. (PHB, 28)
You get a, say 5. It still isn't high, so you choose to use your luck feat.
Whenever you make an attack roll, an ability check, or a saving throw, you can spend one luck point to roll an additional d20... You choose which of the d20s is used for the attack roll, ability check, or saving throw. (PHB, 167)
Second Case: Feat then Feature
Or if you use a luck point and that dice rolls a 1, do you reroll it with Halfling Luck and then still get to choose any?
Not quite. The roll is whichever you choose when you use in the feat Lucky. So, you could choose the 1, and then get to re-roll, as per RAW, but you'd have to use the number of that Halfling luck reroll not the previous two.
Example
So, in this case you roll a 3 and use Luck Feat to roll again, you roll a 1.
Whenever you make an attack roll, an ability check, or a saving throw, you can spend one luck point to roll an additional d20... You choose which of the d20s is used for the attack roll, ability check, or saving throw. (PHB, 167)
You choose the 1, specifically to trigger your halfling ability. So, you've now effectively rolled a 1, and get to reroll it:
When you roll a 1 on an attack roll, ability check, or saving throw, you can reroll the die and must use the new roll. (PHB, 28)
Best Answer
Yes, the Halfling racial feature Lucky works with death saving throws.
Lucky is exactly this, a feature that improves your chances of succeeding on a saving throw.
Even ignoring this (reasonably explicit) statement, a death saving throw is "a special saving throw" - it's pretty clearly a saving throw, and Lucky says that it works on saving throws.