Yes, Edmure surrendered because of threats to his family and because it looked like a no-win situation.
Family plays a huge part in the Game of Thrones universe.
It's important in ours as well, of course, but for them it's more than just loving and protecting your family. It's about duty to your family; honour, respect, legacy, loyalty.
If you don't show proper love, care and loyalty to your own family, why should your other allied houses be loyal, respectful and care for you?
Also the leader of a house without an heir won't be a powerful house for very long.
But most importantly I think it came down to a father having his son threatened. Despite not knowing the child, Edmure no doubt feels a great deal of attachment and sense of protection for his son.
We've seen men go to great lengths to protect sons:
- Eddard Stark protects newborn Jon Snow to great lengths, despite Jon being a bastard. Even though in Season 6 we learn that:
Jon Snow is the son of Eddard's sister Lyanna. Despite not being his son, Eddard is duty and honour-bound to protect this child at all costs.
- Tywin Lannister is disgusted at his newborn son, Tyrion for "being a monster" and "killing" Tywin's wife. Despite this, Tywin explains that he did not throw Tyrion into a river because Tyrion is Tywin's son and he is a Lannister (family).
Jaime threatens that he will catapult Edmure's son over the walls of the castle and then lay siege to it. Knowing that the Lannisters are rich and powerful (at least to Edmure's knowledge), and with the Throne at their side, Edmure sees all of this as a losing situation.
He reasons that the best bet for his survival, his son's survival, his family's legacy and the lives of his people (all of House Tully), is to surrender and live so that he can try to be powerful once again some day.
That's my understanding anyway.