Thanks to Lommy
for reminding me about this topic, for which Fea requested assistance. Just to make things clear in my mind as well:
Bride price = that which is owed by the suitor for the woman's hand in marriage. This is NOT actually "buying the woman", but paying her family for two things: (1) the loss of the good things she was and did as part of the household; (2) her security once widowhood arrives.
Part 2 was typical in pre-modern times because women tended to marry much younger than men, who generally had to establish themselves economically (with a trade or actual wealth) before they could start thinking about marriage, and so the man was quite likely to die before she did.
Scarburg itself cannot be the bride price, since that is given by the king the Eorl of Middle Emnet, and if there is an eldest son, it should go to him, depending upon the king's blessing.
Condition number 1 (above) is rather subjective in Saeryn's case, and that's a matter of agreement between Degas and Eodwine. Condition number 2 has to do with making it the husband's responsibility to care for his wife after his death, rather than making it his childrens' responsibility. Of course, that would mean that Degas' family would be held responsible to take Saeryn back to the Folde, should she wish to go there, since that's where the bride price was paid to. Then again, she could request the bride price be paid to her after Eodwine's death, and she could remain at Scarburg with her children.
The bride price could be coin, or it could be cattle, horses, you name it; could even be a "tithe" of the next five years' harvest in grain, or some such. We could be very creative about it.
Then there's the dowry. The whole idea here is that the family of the bride hopes to enrich the bride and her husband's fresh start by giving them something to bank against the immediate future.
So the dowry has to do with "start-up", and the bride price has more to do with the bride's security in the long term.
Hope that helps.