Instead of a gray-skinned version of the Kurgan, I thought a Monk might be interesting (personality-wise, I’m picturing Groot with more vocabulary). 

Which monastic tradition do you guys feel would work with their abilities?

Goliaths don't really have a racial trait that interacts better with certain Monk subclasses. So really, if you can think of a subclass with lore that fits your character concept then go for it.

I will say though that Way of the Astral Self might be nice. It lets you use Wisdom for Strength checks and saving throws. This is nice because it lets you play a grappling monk without a high strength. And it lets you prioritize wisdom over everything since it increases your AC, saving throw DCs, and attack and damage rolls.

Well, goliaths are a hard, enduring people, so maybe drunken warrior?

Astral Self would work well with goliaths' innate Athletics proficiency, letting them take advantage of using Wisdom for pushing/grappling.

Long Death would work as a tanky Monk, using both the subclass's temporary HP and goliaths' damage reduction.

With their traits giving cold resistance and drawing on unyielding stone for the damage reduction it made me think of Four Elements Monk. Unfortunately 4E has issues mainly stemming from too high Ki cost of their Elemental Disciplines and too limited number of Elemental Disciplines known. But maybe your DM can work with you on this. 

Imagine a half-giant with dragon powers,

wide shimmering, ethereal wings unfurling behind him as he dives into combat from a high rock shelf,

his large fists blazing with elemental might,

his breath a wave of destruction,

and his eyes glowing with draconic power high above other, somewhat lesser men.

Plus, dragons like mountains, and goliaths tend to live up there as well.

Quote from Dalinar2015 >>

Wait wait wait am I getting this right there’s death of old age in dnd

 Yes, but it typically doesn’t come up in my experience 

Shadow monk would be hilarious if you could make the character look like just a big dumb barbarian, then suddenly he just appears behind the enemy who has underestimated him. 

If the idea is to get as far away from the goliath barbarian stereotype as possible, Way of Mercy might be fun

i like making a oxymoron PC like a 'strength ruin knight fairy' but to be honest for a goliath monk i feel like ether Way of the Kensei or Way of the Open Hand might work best depends on ware they grown up or trained from had the means to make weapon's or not. 

If you mistakenly drink a potion of aging because you thought it was a healing potion. My DM done my character way back in the day like this.😀

1) Pick the Goliath with the ability to make a target PRONE

2) Take Sailor background (5' Push with unarmed attack

3)  Take Crusher (5' push with bludgeoning damage)

4)  Attack.  Upper hut Punch damage + Falling damage. (I know just 1D6 damage) And target prone.

5)  Pound away on target.

Take a dip in Ranger, hunters mark, weapon mastery (nick, vex) , Mercy monk. Grappler feat, (Druid/ranger in group - spike growth) Increased movement, go too. Drag the grappled up down the spike growth. Oh the pain and suffering, can’t wait to make this happen.

If you are looking at a 2024 Goliath Monk now, four elements is great for shoving and grappling. As you can do it with reach.

the Grappler feat is good now too and would be a good way to get 18 dexterity at lv4 as it is a half feat

Unless you do what Travis is doing in Critical Role and role a d100 for a really old character after every long rest to see if they died of old age 😂

edit: This was replying to another comment. I’ve never commented on a post on here before lol

Quote from BobbySioux >>

Instead of a gray-skinned version of the Kurgan, I thought a Monk might be interesting (personality-wise, I’m picturing Groot with more vocabulary). 

Which monastic tradition do you guys feel would work with their abilities?

 Depending also what Goliath background also,

misry step x your prof bonus 

or knock prone 

I used the misty step, extra movement, pop up from behind, grapple. I like mercy because of the utilization support to the group. + our healer never heals.