Let $ (x,y) $ be a point on the hyperbola $ x^2 + 8xy + 7y^2 = 225 $. Effectively, we want to minimize $ x^2 + y^2 $. Let $ k = x^2 + y^2 $. Multiplying this with the equation $ x^2 + 8xy + 7y^2 = 225, $ we get \[kx^2 + 8kxy + 7ky^2 = 225x^2 + 225y^2,\]so \[(k - 225) x^2 + 8kxy + (7k - 225) y^2 = 0.\]For the curves $ x^2 + 8xy + 7y^2 = 225 $ and $ x^2 + y^2 = k $ to intersect, we want this quadratic to have a real root, which means its discriminant is nonnegative: \[(8ky)^2 - 4(k - 225)(7k - 225) y^2 \ge 0.\]This simplifies to $ y^2 (36k^2 + 7200k - 202500) \ge 0, $ which factors as \[y^2 (k - 25)(k + 225) \ge 0.\]If $ y = 0, $ then $ x^2 = 225, $ which leads to $ x = \pm 15 $. The distance from the origin to $ P $ is then 15. Otherwise, we must have $ k = x^2 + y^2 \ge 25 $. If $ k = 25, $ then $ -200x^2 + 200xy - 50y^2 = -50(2x - y)^2 = 0, $ so $ y = 2x $. Substituting into $ x^2 + 8xy + 7y^2 = 225, $ we get $ 45x^2 = 225, $ so $ x^2 = 5, $ which implies $ x = \pm \sqrt{5} $. Thus, equality occurs when $ (x,y) = (\sqrt{5}, 2 \sqrt{5}) $ or $ (-\sqrt{5}, -2 \sqrt{5}), $ and the minimum distance from the origin to $ P $ is $ \boxed{5} $.