To find the fraction of the total pressure exerted by hydrogen, we need to use Dalton's Law of Partial Pressures, which states that the total pressure of a gas mixture is the sum of the partial pressures of each component gas in the mixture. The partial pressure exerted by each gas is directly proportional to its mole fraction in the mixture.

First, let's determine the number of moles of methane (CH₄) and hydrogen (H₂) given that equal weights of each are mixed.

The molar mass of CH₄ is approximately 16 g/mol, and the molar mass of H₂ is approximately 2 g/mol. Let the common weight of each gas be $ w $ grams. Then, the number of moles of methane is given by:

$$ n_{CH_4} = \frac{w}{16} \text{ moles} $$

The number of moles of hydrogen is:

$$ n_{H_2} = \frac{w}{2} \text{ moles} $$

Next, we calculate the total number of moles $ n_{total} $ in the mixture:

$$ n_{total} = n_{CH_4} + n_{H_2} = \frac{w}{16} + \frac{w}{2} $$

$$ n_{total} = \frac{w}{16} + \frac{8w}{16} = \frac{9w}{16} \text{ moles} $$

To find the fraction of the total pressure exerted by hydrogen, we calculate the mole fraction of hydrogen, which is the ratio of the number of moles of hydrogen to the total number of moles:

$$ \text{Mole fraction of } H_2 = \frac{n_{H_2}}{n_{total}} = \frac{\frac{w}{2}}{\frac{9w}{16}} = \frac{16}{18} = \frac{8}{9} $$

Dalton's Law implies that the fraction of total pressure exerted by hydrogen is equal to the mole fraction of hydrogen. Therefore, the pressure fraction exerted by hydrogen is:

$$ \frac{8}{9} $$

Thus, the correct answer is Option B: 8/9.