Russ gave you some good advice. But I had a situation in the last year that was similar so I wanted to share my experience.
I have a regular 4 week cut and color client who's husband died suddenly and she didn't tell me for six months. She comes to me to feel good, and she didn't feel like having that conversation again. She wanted her hair appointment to be her escape from what was going on in her life.
It's a different thing for you, since you already know. Just don't be suprised if she steers the conversation away from the topic, or even if she gets upset. She won't be upset with you, she will probably just be sad.
Often when somebody is very strong for a sick family member they are fighting, the person will lose the strength when the loved one passes. Another thing that can happen is the person feels a weird sense of relief coupled with guilt for feeling relieved. It's really deep stuff, and not what someone wants to think about while they get their hair done.
Just be what she needs you to be. Some clients don't really want you to be their friend. For others, being their friend is the most important thing in the world! Just be sensitive to her needs and you will be fine.
I would avoid talking about it in terms of death or loss. Out of respect you might want to say something about it since you know, but don't feel you have to. You didn't mention names, but for the sake of explanation I'll call the cousin Mary and the husband Bill. What I would say is "Mary was in last week and she told me about Bill, I'm really sorry. Is there anything I can do?". Then let her take control of the conversation. She might drop the topic, or she might cry. Don't get freaked out either way.
Or you could avoid it and let her tell you (or not) by starting the conversation with "How are you today?". Let her take the lead. If she starts to talk about it, then you can tell her about the conversation with the cousin. If not, drop it.
Whatever you decide to do, just do what feels right and be natural about it. It will be ok. There is no perfect way to handle a situation like this.