1. **State the problem:** Find the mean of the two surds $\frac{1}{2\sqrt{2} - 1}$ and $\frac{1}{2\sqrt{2} + 1}$.\n\n2. **Recall the formula for the mean:** The mean of two numbers $a$ and $b$ is given by $\frac{a+b}{2}$.\n\n3. **Write the mean expression:** $$\text{Mean} = \frac{\frac{1}{2\sqrt{2} - 1} + \frac{1}{2\sqrt{2} + 1}}{2}$$\n\n4. **Find a common denominator for the sum in the numerator:** $$\frac{1}{2\sqrt{2} - 1} + \frac{1}{2\sqrt{2} + 1} = \frac{(2\sqrt{2} + 1) + (2\sqrt{2} - 1)}{(2\sqrt{2} - 1)(2\sqrt{2} + 1)}$$\n\n5. **Simplify the numerator of the fraction:** $$(2\sqrt{2} + 1) + (2\sqrt{2} - 1) = 2\sqrt{2} + 1 + 2\sqrt{2} - 1 = 4\sqrt{2}$$\n\n6. **Simplify the denominator using difference of squares:** $$(2\sqrt{2} - 1)(2\sqrt{2} + 1) = (2\sqrt{2})^2 - 1^2 = 4 \times 2 - 1 = 8 - 1 = 7$$\n\n7. **So the sum is:** $$\frac{4\sqrt{2}}{7}$$\n\n8. **Now divide by 2 to find the mean:** $$\text{Mean} = \frac{\frac{4\sqrt{2}}{7}}{2} = \frac{4\sqrt{2}}{7} \times \frac{1}{2} = \frac{4\sqrt{2}}{\cancel{7}} \times \frac{1}{\cancel{2}}$$\n$$= \frac{4\sqrt{2}}{14} = \frac{2\sqrt{2}}{7}$$\n\n**Final answer:** $$\boxed{\frac{2\sqrt{2}}{7}}$$