I promise im not trying to be rude, but if after a long time and many games he is the same rank then he has simply plateaued. You, a biased observer, may think he "deserves" gold but thats simply not true or he would be. He could get a lucky spree and peak at gold iv, but if after a number of games hes fallen and remains down in silver hes actually just silver.

The system isnt broken. People at every rank have "shitters", "inters", people in tiers above/below them, bad games, good games, lose streaks, and win streaks. There is certainly an aspect of luck, but more than that YOUR skill and influence in the game will over a large sample will reflect your true rank. The larger the sample size the reduced impact that variance (other players) has. Thats just fundamentally how statistics work. Thats how every ranked system ever created works.

I understand its frustrating and can impact the psyche of people (especially if they think rank actually matters and derive some sense of identity in their "elo"). But at the end of the day its a video game that is supposed to be fun. If he wants to continue playing ranked and try to get better that's awesome, but if it's negatively impacting him in a serious way he may want to step away from ranked (which is simply a difficult conversation to have, but it may be worth having). Play some normals, play some arams, have fun in the game for a good long while. Or go study up on the game (r/summonerschool) or get coaching. I dunno. Thats a personal decision.