First, we need to convince seven of your neighbors of Derb's analysis, to wit, that the probability in question is 13/27. Then we need to find a large number of fathers with two children, at least one of whom is a boy. It doesn't matter on what days they were born. Have them bring the birth certificates. I hope you will agree with me that the probability that any father in this group has two boys is 1/3 (or 0.333, not 0.48).
Then we segregate the fathers into seven groups according to what day of the week the boy was born on. If the father has two boys born on different days of the week, have him flip a coin and join son #1's group if heads, son #2's group if tails.
Have the group with a boy born on Tuesday line up, and then repeatedly knock on neighbor #1's door and say:
"I have two children. At least one is a boy. He was born on Tuesday. Two dollars will get you three if I have two boys."
Hopefully neighbor #1 will take the bet, having been convinced that he's getting good odds. He's getting a 3:2 payout on a bet that he thinks is approximately 1:1.
Have the group with a boy born on Wednesday line up, and then repeatedly knock on neighbor #2's door and say:
"I have two children. At least one is a boy. He was born on Wednesday. Two dollars will get you three if I have two boys."
Do this with the groups for the remaining days of the week and remaining neighbors. As far as we are concerned, we are paying out at 3:2 a bet that is actually 2:1! (A fair payout would be four dollars for every two dollars bet. We are paying out as though the odds were 2/5 or 0.40. Our advantage is 7%, better than the house advantage in a typical casino game. )
As far as any single neighbor is concerned, he's simply seen a repetition of the Tuesday Child problem. Everybody who knocks on his door says the same day of the week. Once you've got all their money, they might get suspicious, but you've got the birth certificates to back it up... and the (bogus) mathematical analysis. Some people are just lucky, you will tell them.