The name is Alexander Erik Hubertus Bertil.
Alexander is not a traditional upper-class let alone royal name in Sweden, rather it has some white trash connotations (like so many other typically Scottish names made famous in Scandinavia by Hollywood: Kenneth, Kenny, Kevin, Roy, Glen etc.) It was thus the expected choice from the prince's reality TV starlet mother, who, for all we know, may have chosen it because of some sentimental souvenir from a fling with an Alexander, like Paradise Hotel moderator Alex(ander) Schulman.
NB it should be noted that in Finland, also among the Swedophones, Aleksander is a much more accepted name in all layers of society, because of the Russian era. Example: Current Finnish minister of foreign affairs Aleksander Stubb. For the same reason Alexander also has a somewhat Russian connotation in Sweden.
PC-obsessed Swedes (and there is a lot of them!) are barking about the prince being named after the King's (Carl Gustaf Folke Hubertus's) Nazi uncle, Prince Hubertus of Saxe-Coburg-Gotha, a soldier who was killed in action in Ukraina in 1943. Probably they haven't bothered to force their stupid little PC brains around the fact that he also was gay.
Many had expected the prince to be made Duke of Dalarna, his mother's native province, something which the Dalecarliens apparantly also wished. But he was not. Maybe because it was too un-Swedish to have the Duke's grandparents live in his duchy as some sort of semi-nobility, but more likely he was made Duke of Södermanland because he is to inherit the lease of Stenhammer castle and estate in Södermanland, left to a male Bernadotte, with preference for a Duke of Södermanland for 1000 Swedish crowns annually by a childless marshall of the court, Baron Robert von Kræmer, in 1903. It was quite a sum in 1903, but a bargain today. Untill his death in 1965 Prince Wilhelm, Duke of Södermanland, was the beneficiary of this lease.
As far as I can see there has only been one (semi-)Swedish royal called Alexander earlier: 17th-century Prince Alexander Karol Vasa, son of King Sigismund Vasa of Poland and Sweden.
Relationship to ancestors of Prince Alexander, Duke of Södermanland, with the name Alexander:
1. Prince ALEXANDER of Sweden
2. Prince Carl Philip of Sweden
3. King Carl XVI Gustaf of Sweden
2. Prince Gustaf Adolf
4. King Gustaf VI Adolf of Sweden
5. King Gustav V of Sweden
6. King Oscar II of Sweden
7. Josephine of Leuchtenberg, Queen of Sweden
8. Eugène de Beauharnais, Duke of Leuchtenberg
9. Vicomte ALEXANDRE de Beauharnais
1. Prince ALEXANDER of Sweden
2. Prince Carl-Philip of Sweden
3. King Carl XVI Gustaf of Sweden
4. Prince Gustaf Adolf of Sweden
5. Victoria of Baden, Queen of Sweden
6. Louise of Prussia, Grand Duchess of Baden
7. Augusta of Saxe-Weimar-Eisenach, German Empress
8. Grand-Duchess Marie Pavlovna of Russia, Grand Duchess of Saxe-Weimar-Eisenach (sister of Emperor Alexander I and mother of Grand Duke Carl Alexander)
9. Paul I, Emperor of Russia
10. Peter III of Holstein-Gottorp, Emperor of Russia
11. Grand Duchess Anna Petrovna, Duchess of Holstein-Gottorp
12. Peter the Great, Emperor of Russia
13. Alexis I , Tsar of Russia
14. Michael I, Tsar de Russia
15. Fyodor Nikitich Romanov, Patriarch Philaret of Moscow
16. Princess Evdokiya Alexandrovna Gorbataya-Shuyskaya
17. Prince ALEXANDER Borisovich Gorbatiy-Shuyskiy, died in 1565, famous general of Ivan the Terrible, conqueror of Kazan.
The Princes Gorbatiy-Shuyskiy are descended from Andrey Yaroslavich, Grand Prince of Vladimir and Prince of Suzdal, brother of ALEXANDER nicknamed Nevsky, who defeated the Swedes on the banks of the Neva. It is possible that they and the Romanov also were descended from one of Alexander Nevsky's sons, but I can not find the link.