Information on the son of Väinämöinen is very patched, so I'll cover multiple instances here.
First, let's see the very little that remains of the circumstances of his birth.
  Alina hyvä emäntä Löysi lapsen lastusista Isä pani Ilmariehtaroille Vel' vennon joukariksi Sisar sotijaloksi Emä pani pakkaisiksi Sinä vanha Väinämöinen Kuin teit huorin kolmiöissä! |
Alina, good lady Found a child from shavings Father named him Ilmariehtaro Brother, Joukari of Vento Sister Sotijalo Mother named him Frost You old Väinämöinen Committed adultery at three nights old! |
Makes no sense, yeah. But based on Karelian runosongs, we can figure out the plot: A child is found but cannot be named because its father is not known, and names must come from ancestors. Different kinds of names are tried to give to the child, but none "stick" (they are not spiritually right for the child). Väinämöinen as a judge states that the fatherless child should be taken to the swamp and struck dead with a piece of wood. Magically, the three-night-old child begins to speak and asks why he should be killed when Väinämöinen hasn't been punished either despite his wrongdoings, including adultery as it is Väinämöinen himself who is the father of the child. Väinämöinen had copulated with a water maiden, a woman by the strait.
The story also morphed into a Christian ballad where a man named Anterus or Hannu impregnates a woman named Margareta or Marketta. She gives birth to the child and hides it in shavings (abandons it?) but Helena (namesake of Saint Helena) finds it. No man admits to being the child's father, until it gains the ability to speak magically and reveals the truth to everyone.
In different runosongs, he is connected to both Frost and Iron, partially because he can also stop bleedings caused by iron. Above, we can already see one frost connection.
 Pakkanen puhurin poika Jos olet sitä sukua Kuin on enne Osman poika Pohjanmuan kylästä Kotoisin Vänkämöinen Älä kylmä kynsiäin Äläkä kohmo kopriain Kylmä kylmiä kiviä Jäätä järven rantasia! |
Frost, son of gale If you are of the family As once was the son of Osma From a village in Ostrobothnia Vänkämöinen was from Do not freeze my nails And don't numb my hands Freeze cold stones Put ice on the coasts of lakes! |
One Ladoga Karelian runosong also calls Frost the same as the secret son of Väinämöinen.
Rauta poika Vuolangoinen/Vuolamoinen Vuolan tytterän tekämä Sukunna vanha Väinämöisen! |
Iron, son Vuolangoinen/Vuolamoinen Created by a daughter of Vuola Of the family of old Väinämöinen! |
The Daughter of Vuola is also called Vuolahatar. One Ladoga Karelian runosong gives the secret son the name Vuolervoinen. In some instances, it's not that iron is Vuolahainen himself, but that he is the son of Vuolahainen. If we follow Kaarle Krohn's logic that this word ultimately comes from "Vuojolainen", it would essentially be synonymous with someone from Pohjola. Then Rauta poika vuojolainen "Iron, son from Vuojola".
Painajainen huoripoika Tuulijaisen tuuvittama Itse vanhan Väinämöisen! |
Nightmare, bastard son Lulled by Tuulijainen Of old Väinämöinen himself! |
Here, he is listed as the helper of the black man from the sea.
| Mies musta merestä on Uros aallosta yleni Neljän peukalon pituinen Kolmen sormen korkulainen Parta rinnalla etessä Hiukset kannalla takana Jalassa kiviset kengät Piässä rautainen kypärä Ei enne ero tultu Salapoika Väinämöinen Nosti miekase merestä Lapionsa lainehestä Jolla ruohtais rupia Jolla painoi paisumia |
A black man is from the sea A hero rose from a wave His height four thumbs Three fingers A beard was on his chest on the front Hair on his heels in the back Stone shoes in his feet Iron helmet on his head And no peace came before Secret son Väinämöinen Lifted his sword from the sea His shovel from the wave With which he'd unroot scabs With which he pushed down abscesses |
The son, too, is described to be a tietäjä.
 Itse vanha Väinämöinen Tultaki tupessa kanto Siitä iski ilman alle Vaimolleen valkeata (Se oli sitte vaimo pessyt lasta ja keittänä vettä) Vielä vettä lämmitteli Pesipähän pientä Poikalasta pyöritteli Kanteli kapalolasta (Siitä tuli mies sitte. Ei muista millä lailla kasvatettu.) Siitä kasvo kaunis lapsi Uros aina oivallinen (Ja siitähän oli loihtija sitten tullut Väinämöisen pojasta. Se on lähtennä sitte, niin on saanut kalaa) Ja tehnyt luikun lehmän luista Härän sarvista hälinän Kuhtu kontiot kotihin Ja karhut keskikartanolle Se on niitä tappanna ja saanut suuria kaloja järvista, jos kuin paljo mahilla. Loihtinna vielä: Noita ennen eukko neuvo Oma vanhempi opetti Omat on saamani sanani Omat tieltä tempomani Risukosta riipomani Kanervikosta katkomani |
Old Väinämöinen himself Carried fire in his sheath as well Struck it under the sky To get fire for his wife (The wife then bathed the child and boiled water) Warmed up some water still Bathed the little one Turned around the son Carried around the swaddled child (He became a man then. I don't remember how he was raised.) There, a beautiful child grew up The most excellent hero (And the son of Väinämöinen sure had become a sorcerer. He had gone then and got fish) And made a horn from the bones of a cow A jingling thing from the horns of an ox Invited bears home And bears to the middle of the yard He had killed them and got large fish from the lakes, so many with his might. He had also spellcasted: My old mother once adviced me those My own parent taught me The words I got are my own Snatched by myself on the road Ripped from a thicket Cut from the heather |
Funny enough, one Savonian version turned so that Väinämöinen married Louhi and they had a son, and when he was fifteen, Louhi sent him to get her inheritance from the north, in what is a very warped version of the Theft of Sammas. The son has to defeat northerners with shamanistic abilities just like Lemminkäinen in his myths.