Honestly, the best way to introduce money, the idea of money, into a barter economy is not with logical reasons and coaxing but with stories.
There's a children's story, or maybeso a set of stories, where the main character wants to trade for something, and has to go on a long quest to get from someone who wants the thing he has, to one who can give him the thing he wants, because of all the people between who are picky about what they want for what they have. Stories like this might catch on pretty well in a barter economy, they are familiar with the web of local favors and debts - but it will also subtly lay the groundwork that the chain of bartering is difficult and inefficient.
Then, your character may have stories or tales or anecdotes (either their own set of stories, or anecdotes about their native land) about the use of money, to circumvent this. Instead of needing to visit fifteen or twenty people to make a trade chain to get what someone needs, wouldn't it be easier to have tokens that will let you give what you have for them, and get what you need for them? The idea of using tokens to keep track of debts might or might not make immediate headway, depending on the size of the population and how good the system they already have is, but the idea will be out there for people to consider.
As a bonus, you say the tribe is at a stage where they're settling from nomadic to farming. That means their ways are changing, they're figuring out different ways of doing things that work, and letting go of ways that don't. Their population will be growing, and they might start running up against the kinds of problems that make money easier than barter, like flexibility or not having to keep track of too many people or debts. They are already changing, your person just needs to make sure the idea is out there and let them figure out if they think it will work better than what they have.
If the idea is there, even in stories, it will be on the radar and available for when the society starts looking for solutions to the problems a barter economy might have in a community becoming larger, more specialized, and less flexible. Especially if the person from the money-culture uses the stories to frame the idea as something that helps solve those problems.
This way, the society may be convinced to try and work out a system that will work for them, based on their own thinking and ways of dealing - inspired by a story, but their interpretation of it, instead of trying to impose or coax the society to adopting an system very much outside their culture. Maybe they won't adopt your person's culture's use of money directly, but with the idea that it can be done differently than plain barter in their minds, they can think about, and talk about, how to make their own system. Maybe instead of centrally backed money (like a bank's) they will pick a common trade good for "pricing" (gold, salt, cocoa beans), or they will start with individual tokens for debts and go from there. It will eventually evolve to suit their culture and needs, but the way it does so will be organic, and therefore a lot more likely to be successfully adopted.