The transaction costs of uprooting yourself to obtain new housing are particularly high, relative to other goods. This is not the same as making a choice among buying from different vendors of apples, or electronics.
But in the situation like the one Person A faces, with high transaction costs, there is the opportunity for an innovator to create value for Person A, and profit from it. Lets assume that someone else, Person B, were able to invent an amazing new moving van that could reduce Person A's transaction costs of moving to $500. Then even with the smaller, $50 rent increase, it would be worthwhile for Person A to move. Person B could then charge person A a certain price, (less than the $100 difference between the new transaction costs and the rent increase,) and both Person A and B would benefit.
The facilitation of transactions that would otherwise not happen is one important way that innovators can create value for society. For example websites make it easier to shop around for housing, which reduces the opportunity costs for housing seekers. (The same goes for Star Trek collectibles.)
Increasing benefits and reducing costs is the essence of innovation. And in the end, innovation is the only reliable way for individuals to get abnormally rich. They must find ways to abnormally reduce the costs or increase the benefits for others, and demand a price for it. Finding ways to allow mutually beneficial trades to happen, that would not otherwise happen due to large transaction costs, is one way to achieve this goal and make a buck.