A letter marked refused can mean one of two things, and figuring out which applies here matters before drawing conclusions.
The first possibility is that your cousin actively refused the letter. Inmates do have the right to reject incoming mail without opening it. If there are unresolved feelings, embarrassment about the incarceration, or a desire to cut off contact with certain people during this period, refusing mail is one of the few things entirely within their control. A recent incarceration in particular can produce this kind of reaction. Some people need time to process where they are before they are ready to engage with people from their outside life, even family members they care about.
The second possibility is an administrative issue rather than a deliberate choice. If the address was formatted incorrectly, the inmate's name did not match what is in the facility's system, or the letter arrived during a period when your cousin was being transferred or had just arrived and was not yet fully processed into the system, the facility may have returned it without it ever reaching him.
The way to distinguish between the two is to contact the facility directly and ask about the return. A counselor or the mailroom staff can often tell you whether the refusal was inmate-initiated or administrative. That answer shapes what you do next.
If it was administrative, correcting the address or name formatting and resending through InmateAid resolves it. If it was your cousin's choice, giving it some time and trying again with a brief, low-pressure note may be the right approach. Not everyone is ready to receive contact immediately after arriving inside.