I'm gonna have to say that Hinata's surname will remain as it is. Because, if we take the case of two other couples, who were from different clans and had offsprings, we can see that wives' surname remains as it is. Examples are:
- Minato Namikaze and Kushina Uzumaki
- Asuma Sarutobi and Kurenai Yūhi
In both cases, even after their marriage, their surnames were kept the same. So this leads me to believe that Hinata will retain her surname as Hyuuga.
Taking the same examples for the case of the offsprings,
- Mirai Sarutobi
Mirai's case, she retains her father's surname. While Naruto's official surname is Uzumaki. But the Uzumaki surname was given to him because it was a means to protect him and his identity (from enemies). I think there was a chapter which discussed that incident. Therefore, his original surname was Namikaze.
From these examples, I think its safe to say that the child recieves the surname of the male parent. Which means the names will be Boruto Uzumaki and Himawari Uzumaki.
Update: (credits to @AyaseEri)
As for physical evidence, the 8th official guidebook, Zai no Sho (released Aug 7th 2015) confirms that Boruto and Himawari have their surname as Uzumaki:
Boruto Uzumaki (page 26)
Himawari Uzumaki (page 31)
But it seems my logic with the wife's surname being preserved is faulty. As in Zai no Sho, Hinata's surname is not Hyuga, but Uzumaki (page 31). Therefore, her official name after marriage has become Hinata Uzumaki.