Just about every game I've ever been in (player and GM), players deliberately wrote open-ended hooks into their characters' backstories like this in the hope that the GM would integrate them into the story and at some point surprise the players with plot related to their backstory. Not every such hook can come up (at least not in a short or medium length campaign), but when they do it's a great way to add emotional depth to a story and share meaningful parts of your character's history with the group.
As an example, one of my absolute favourite gaming experiences came when I wrote up a character who's backstory included having been in a car accident that put his daughter in a coma; he later bargained with a reality altering demigod to change history so that it didn't happen, and the new history had my character and his daughter escape unscathed while the driver of the other car now died in the accident. That was years-old backstory at the start of the game. Many months of real world time later, the GM ran a "side story" where we played new characters he wrote up for us. It turned out the new character I was playing was the other driver I had "killed" in writing my character's traumatic backstory (and the whole side-story was a vision that the regular characters were experiencing). Suddenly I knew and cared about the life I had unknowingly traded for my daughter's, which sparked a whole side-quest to atone for it. Much pathos. I had no idea that such a thing would happen, but I wrote details like that into my backstory specifically so that if the GM thought of something cool they could use it.
The fact that the player wrote that their character never actually knew they brother which they're searching for suggests to me that the player is thinking this way; that they are deliberately giving you freedom to decide what the brother is actually like, so that you can integrate this quest into your storylines in whatever way you think best.
Since this is pretty conventional in my gaming circle, I'd probably just take my idea and run with it if I was the GM. But there's a way to be sure: ask the player what their intent was in writing that bit of backstory.
Are they really giving you a plot hook to see if it inspires you to write story related to their character? If so you can do whatever you think would be most interesting (preferably in a way that is interesting for the other players too, so you get everyone invested), and not tell the player so it will be a surprise (although perhaps not a surprise that it's coming now).
Do they have specific ideas about where they want that story to go, or what they want the brother to be like? If so you should definitely have a chat with the player about what those ideas are so you can work with them, but it should be a discussion; you (presumably) know more about where you're planning to take the game than the player does, and don't feel obligated to write in anything and everything your player demands (especially since you're the one who's going to have to act out the NPC).
Or was it just intended for a bit of backstory colour, and they don't actually want it to come up in game at all? Again, it's your game to run, not the player's, but you usually don't really want to put a lot of effort into making a story that ties in your players' backstory only to find out they don't actually want to explore that at all (can be especially problematic since they might feel that their character should want to explore it, even if the player doesn't, so you might not get an accurate read on that just by observing how they play).