Perhaps a coin-flip should be used in this case. Overall, this proposal is the best modification to the English surname convention that I have been able to think of, and I recommend it. Variations Anna Diller suggests that sons should inherit their surnames from their mothers, and daughters from their fathers, in order to counteract men's preference for sons. That might be useful, but would make the new convention less backward-compatible with the existing convention and less intuitive, and therefore harder to get people to agree on. Everyone has two surnames.
Children take their mother's first surname and their father's first surname, in either order.
Neither spouse's name changes when they marry. Nuclear families are referred to by both parents' first surnames. Notice that if sons were to put their father's surname name first and daughters were to put their mother's surname first, then the first surnames would match the ones from my proposal. However, I suspect that most families use the same ordering for all the children.