Before reading the comments I voted for 1/3, but now I'm convinced of 13/27.

I had to write a Python program to confirm it, but I'm now a 13/27 believer. See for yourself (swap the commented lines with the ones above them for the version without "born on a Monday"):

import random

cases = 0
successes = 0
for i in range(1000000):
  g1 = random.randint(0, 1)
  w1 = random.randint(0, 6)
  g2 = random.randint(0, 1)
  w2 = random.randint(0, 6)
  if ((g1 == 0 and w1 == 0) or (g2 == 0 and w2 == 0)):
  # if (g1 == 0 or g2 == 0):
    cases += 1
    if (g1 == 0 and g2 == 0):
      successes += 1
  else:
    continue

print(successes / cases)
print(13 / 27)
# print(1 / 3)

The best way I can intuit this is that by specifying that one boy is born on a Monday, the likelihood that the boys are interchangeable vastly decreases.

I guess that raises the interesting question of specifying something less definite, like "at least 1 kid is a boy who likes ice cream". Even without knowing the exact proportion of boys who like ice cream, the fact that there are some boys who don't like ice cream bounds the conditional probability to be strictly greater than 1/3.

There are 14x14 = 196 possible "gender_day" pairs. For example "girl_tuesday/boy_friday", "boy_sunday/boy_wednesday", etc. If at least one child is a boy born on a Monday, then we've eliminated all pairs of the form "girl_X/girl_W", "girl_X/boy_Z", "boy_Z/girl_X" and "boy_Z/boy_Z" (where X, W are days and Z is a day that's not Monday). So that means we've eliminated 7x7 + 7 x 6 + 6 x 7 + 6 * 6 = 169 cases. There are therefore 196 - 169 = 27 cases in the set of possibilities. In this set of possiblities, the families with 2 boys are of the form "boy_monday/boy_X" and "boy_X/boy_monday", of which there are 7 + 7 - 1 = 13 (we subtract 1 to avoid double counting the "boy_monday/boy_monday" case). So that gives us 13/27 as the final probability.

There are 27 cases: 13 where only the older kid is a boy born on a Monday (6 where the younger is a boy born on Tuesday-Sunday, 7 where the younger is a girl born any day of the week), 13 where only the younger kid is a boy born on a Monday (6 where the older is a boy, 7 where the older is a girl), and 1 where both are boys born on a Monday. So, in 13 of 27 cases, both are boys.

Can someone explain why the "on monday" changes anything?

Assume I except the argument that the answer is 1/3 without the "on monday"

@JoshuaPhillipsLivingReaso Where does 1/3 come from? Wait nvm it's because the question implies we don't know which kid is the definite boy

@JoshuaPhillipsLivingReaso I guess because families with two boys are more likely to have at least one of them born on a Monday? So the odds should increase slightly

@JoshuaPhillipsLivingReaso you have to count all the cases for it to make sense. It's just one of those counterintuitive things in statistics

Slightly over 1/3, because we don't know which is the definite boy, and the possibility of identical twins shifts the independent event probability to slightly over the usually assumed 1/2