One of my player want to play Paladin, after he review the Paladin's feature he ask me, "Can we change the damage type of Divine Smite to Cold damage? I worshiping the God of Winter, it should give me cold type damage rather than radiant".

Since there isn't any other smite spell having cold or ice related, I can't give the option to use spell instead.
PS: I know then gods are divine (celestial) being, they should have radiant power, but some gods are related more to a specific element or nature rather than radiant. This have been discussed with my player. 

Currently I only allow him to flavoring the attack as ice coating his sword when he attack, but the damage still radiant. My player seem a little bit disappointed, but he still play the character.

What you guys think about changing the damage type for Divine Smite to fit the character?

Think it would be fine. Radiant is a damage type not too commonly resisted, so changing it to any damage type excluding force damage would perhaps even be a set-back, but it's up to the player. In essence, Divine Smite is a smite spell, you expend a spell slot and use it on an attack, and so long as the player can't change the damage type on the fly it seems pretty balanced. Imo it has always seemed strange to me that Oathbreakers don't deal necrotic instead of radiant. Because, I mean, they're supposed to be the evil paladins. Also remember that paladins don't necessarily have to be sworn to a god, their power comes from an Oath, but that doesn't really matter here.

Quote from PietyTheCharismatic >>

Think it would be fine. Radiant is a damage type not too commonly resisted, so changing it to any damage type excluding force damage would perhaps even be a set-back, but it's up to the player. In essence, Divine Smite is a smite spell, you expend a spell slot and use it on an attack, and so long as the player can't change the damage type on the fly it seems pretty balanced. Imo it has always seemed strange to me that Oathbreakers don't deal necrotic instead of radiant. Because, I mean, they're supposed to be the evil paladins. Also remember that paladins don't necessarily have to be sworn to a god, their power comes from an Oath, but that doesn't really matter here.

 Thanks for the answer, I will use it in the next session with my player. 

I know that Paladins power is not from the gods, but from the Oath they taken.
But my player made his character backstory where the character climb the mountain just to ask for the god the power to protect his allies, later the character hear some voice and make an Oath with the God of Winter and the god give him the power to protect people.

Another option is to alter Searing Smite (fire) to Biting Smite (cold). I agree with not changing the damage type from radiant. The flavor is in the look and feel, radiant could be pulled as freezing energy because it is pulled from a plane of pure radiant energy which is never really given a description. It is not hot or cold, good or evil, heavy or light, it usually manifests as light vs. necrotic which is dark. You could rule that it shapes itself as an expression of the character's soul.

Quote from theHumanChris >>

Another option is to alter Searing Smite (fire) to Biting Smite (cold). I agree with not changing the damage type from radiant. The flavor is in the look and feel, radiant could be pulled as freezing energy because it is pulled from a plane of pure radiant energy which is never really given a description. It is not hot or cold, good or evil, heavy or light, it usually manifests as light vs. necrotic which is dark. You could rule that it shapes itself as an expression of the character's soul.

 Thanks for the advice.

Currently i flavor my player smite as ice coating his sword when he use Divine Smite. Later I want to test if changing the damage type and see how it goes.

To be honest, if you are not throwing anything at them where cold damage is an issue, then it is not an issue. Also, you could just flavor the smite as cold damage that defeats immunity and resistance as a gift from their god and make it special to their character. Their are plenty of abilities that allow a certain damage type to bypass resistance like the Elemental Adept feat, so allowing it to bypass immunity would not be too hard of a stretch for a god, haha. It is all flavor in the end and making the experience for the player as fun and memorable as possible without causing some rules-lawyer hubbub at the table over RAW.  I do enjoy the idea of unique smite energy depending on your god. That would add a little extra role-play sauce to paladins that live by an oath AND/OR decide to swear fealty to a particular god.

Quote from theHumanChris >>

To be honest, if you are not throwing anything at them where cold damage is an issue, then it is not an issue. Also, you could just flavor the smite as cold damage that defeats immunity and resistance as a gift from their god and make it special to their character. Their are plenty of abilities that allow a certain damage type to bypass resistance like the Elemental Adept feat, so allowing it to bypass immunity would not be too hard of a stretch for a god, haha. It is all flavor in the end and making the experience for the player as fun and memorable as possible without causing some rules-lawyer hubbub at the table over RAW.  I do enjoy the idea of unique smite energy depending on your god. That would add a little extra role-play sauce to paladins that live by an oath AND/OR decide to swear fealty to a particular god.

 Wow, i never thought about it, that can work in the game session, thanks for the tips bro. 

I'm a little late to the conversation but there is half of the conversation that is a bit lacking. It's not just who resists cold and is vulnerable to cold...it's also who resists radiant and is vulnerable to radiant (or reacts in a certain way to radiant). A winter paladin dealing cold damage wouldn't technically stop the regeneration of a vampire for example. They would slow down a water elemental though.

To be honest the word radiant gets in the way. It's very difficult to visualize outside of literal damaging light spells (some of which don't sound like they'd even do damage). And then there's spells like sacred flame that maybe look like flame but deal radiant. At least Flame Strike is half fire. I would have called it holy damage or divine damage personally.

As for a paladin dealing cold damage, I'd allow it. I might even allow a choice, especially since I dislike the actual "smiting" spells.

How about changing to Thunder Damage?

OP was asking for a player with a specific idea of worshiping a deity of winter and cold which is why we have been discussing that element mostly. As it goes, a DM has full privileged to alter anything in the game and allowing a concept to drive the narrative is perfect. Just checking on the balance of power for mechanics and unintentionally undermining the character.

Thunder damage could generate a Thor styled paladin, instead of needing to go with a tempest cleric.

Cold is demonstrably a worse (less desirable) damage type than radiant.  I would totally allow them to downgrade their damage from Radiant to Cold.

Reasons why Cold is worse than than radiant.

Monster Manual:   46 have resistance to cold vs 4 to radiant; 20 are immune to cold, vs 0 immune to radiant, and 4 are vulnerable to cold vs 2 to radiant

Spells:  Absorb Energy, Protection from Energy, and Fire Shield all provide resistance to Cold, I can't think of any that protect against Radiant.