trying to create an ice sorcerer, balanced homebrew recommendations please? thx

I know that this may not be what you're asking for, but if you want to create a Sorcerer focused on ice or Cold damage, I've got something to pair with the Homebrew. If you don't make a Homebrew subclass about ice(Which as long as it's balanced, I'd heavily recommend it.), you would want to pick the Draconic Bloodline subclass, and pick the white dragon for the elemental properties. This will give you resistance to Cold damage, and at 6th level you apply your Charisma modifier to all Cold damage you deal, making you a good blaster-caster. For Metamagic, I'd pick Transmuted Spell and Quickened Spell. I'd also recommend getting the Metamagic Adept Feat, either as playing a Variant Human, or trading it out for a ASI, so you can get spells like Twinned and Empowered Spell. This will allow you to cast several spells in one turn, and using Transmuted Spell you can turn it into Cold damage, so you can add to your damage output. For spells, I'd say Ice Knife, Chromatic Orb, Cone of Cold, Wall of Ice, or Prismatic Spells for Cold damage. You could also get high damaging spells, like Fireball, to turn them into Cold damage and optimize your damage output. Still, I'd recommend adding some of your own Homebrew to counteract the build I made, and make it so you don't have to follow this relatively specific build.

yeah, thx for the recs, but i would prefer homebrew, also, to crycket, this is for a multiclass, and my other main types of damage are piercing/slashing, and fire and psychic (because I use illusions)

also, didn't know you could change a spells damage type

Elemental Adept is a really good feat : You ignore Cold Resistance and turn 1 into 2 on damage dice, which is a nice bonus.

1. Frozen Bloodline:

   • This feature represents your magical heritage, tracing back to ancient ice-wielding sorcerers. Mechanically, it grants you access to ice-themed spells and abilities.
   • Implementation:
     • You gain proficiency in Arcana (to reflect your magical lineage).
     • Flavor-wise, you might have frosty blue veins visible beneath your skin, especially when casting spells.

2. Icy Cantrips:

   • These are the basic spells you can cast at will without expending spell slots.
   • Frostbite:
     • A 1st-level cantrip that targets a creature within 60 feet.
     • The target makes a Constitution saving throw or takes cold damage and has disadvantage on its next attack roll.
   • Ray of Frost:
     • Another 1st-level cantrip that fires a beam of icy energy.
     • It deals cold damage and reduces the target’s speed by 10 feet until the start of your next turn.

3. Frozen Heart:

   • At 1st level, you gain resistance to cold damage.
   • You can spend 1 sorcery point to grant yourself immunity to cold damage for 1 minute.

4. Glacial Metamagic:

   • These metamagic options enhance your spells.
   • Flash Freeze:
     • Spend 2 sorcery points when casting a cold damage spell.
     • Freeze the ground around the target, restraining creatures in the area.
   • Icicle Surge:
     • Spend 1 sorcery point when targeting a single creature.
     • Add extra cold damage equal to your Charisma modifier.
   • Implementation:
     • Keep track of sorcery points and apply these effects during spellcasting.

5. Frozen Form:

   • At 6th level, you can transform into a semi-solid ice form.
   • While in this form:
     • You have resistance to physical damage.
     • You are vulnerable to fire damage.
     • You can maintain this form for a number of minutes equal to your sorcerer level.

6. Blizzard Mastery:

   • Starting at 14th level, you can create a localized blizzard.
   • Spend 5 sorcery points to summon the storm:
     • Creatures within a 30-foot radius make a Constitution saving throw.
     • Failed saves result in cold damage and blindness for 1 minute.

7. Absolute Zero:

   • At 18th level, you reach the pinnacle of your icy powers.
   • Once per long rest, unleash the Absolute Zero effect:
     • Choose a point within 60 feet.
     • Everything in a 30-foot radius centered on that point freezes instantly.
     • Creatures caught in the area must make a Constitution saving throw or take massive cold damage and be paralyzed for 1 minute.

Sorcerer blood lines are based on the idea of a magical heritage so you will need to have an idea of a kind of ice thing to be the origin of the sorcerers magic.

For example the storm sorcerer does elementals of air  and being born under a storm. You could make a storm sorcerer variant that is the something like  The storm sorcerer of the blizzard and for that you could make all kinds of blizzard based abilities.  like 

• Shroud of snow, when you cast a spell you can shroud yourself in a veil of snow. Immediately before or after you cast a spell of 1st level or higher you can use a bonus action to become heavily obscured until the end of your next turn 
• Heart of the blizzard. Gain resistance to cold and fire damage. Whenever you start casting a spell of 1st level or higher that deals cold damage, the area around you freezes.  Creatures of your choice that you can see within 10 feet of you to take cold damage (choose each time this ability activates) equal to half your sorcerer level and their movement and their movement speed is reduced by 10ft until the start of your next turn. 

ect...

Other options include frost giants or snow hags in which case look at the monsters for ideas

You could use the Pyromancer Sorcerous Origin from Plane Shift: Kaladesh (see pg. 9), and simply replace all of the fire damage in its features with cold damage.