What does "Our way is the old way" mean?

Question

It is a quote from Bran in Game of Thrones. It is used in this context:

"You understand why I did it?"
"Jon said he was a deserter."
"But do you understand why I had to kill him?"
"Our way is the old way."
"The man who passes the sentence should swing the sword."

Answer 1

Observe the context and the underlying story:

• Bran is witnessing an execution.
• Ned is executing a man who has deserted the Night's Watch.
• The Night's Watch is a group that has sworn to safeguard humans from White Walkers. Those who join the Night's Watch, swear an oath to do so for the rest of their lives. Desertion is punishable by death.
• The Night's Watch has existed for thousands of years.
• Though I don't think it has been explicitly said, it's fair to assume that punishing desertion by execution has been the rule for as long as the Night's Watch has existed. At the very least, no one has ever argued that it is a recent tradition.

Bran's first guess is that the execution takes place because it has been a tradition to execute deserters. It's a rule that has been applied for many centuries, and should remain in active use.

Knowing this, let me translate Bran's words in the script:

Ned -- You understand why I did it?
Bran - Jon said he was a deserter.
Ned -- But do you understand why I had to kill him?
Bran - We follow the rules that have been followed for a long time already?
Ned -- The man who passes the sentence should swing the sword.

Answer 2

There are indications throughout the "A Song of Ice and Fire" series that the people in the North follow practices that have existed in Westeros for longer than the practices followed in the rest of Westeros. The Northern form of life is part of an unbroken tradition that started with "the First Men", who were the first human migrants to Westeros. The most obvious example of this is the religion of the North; Northerners follow "the Old Gods", as opposed to the religiion of the Seven, which was introduced by a group of people who migrated to Westeros later ("the Andals").

When Bran says "Our ways are the old ways", he is saying that we, in the North, follow the old practices of the First Men. Presumably, these old practices include the practice of the man who passes the sentence being the man who swings the sword.