Need to look at it differently. It isn't Fame, it's Reputation. If you search the wiki for "fame", it redirects you to Reputation.
Just meeting and greeting people allows a lot of people to get to know your name (fame), but helping people out a lot builds your Reputation. THAT is what you're trying to do--make people talk good about you and build up your reputation. Look at the dialogue tracks as your reputation builds and it makes more sense:
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...Oh, excuse me. I was just counting my cruor. <Name>? Never heard of you. If you want to make yourself useful, why don't you help that chef girl with her bean stew? I've worked up an awful-wawful appetite.
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<Name>...? I might have heard that name somewhere. He hasn't done anything for me, though, so I promptly-womptly forgotaru it!
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So you're <Name>. Yes, I've heard the occasional rumor aboutaru you. You'll have to keep working hard, though, if you want to become as wealthy--I mean, as famous--as me!
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Ah, <Name>. Your generous deeds have won you quite a few fans around camp these days. When this camp prosper-wospers, so too do my fortunes. For all our sakes, keep up the good work!
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Well, if it isn't <Name>. You've made quite a name for yourself, haven't you? Don't you feel lucky that I picky-wicked you out of the scrap heap and set you on your way to stardom?
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Well, look who we have here! None other than the illustrious-wustrious <Name>, hero of the desert! Your mere presence does this humble merchantaru an honor, I would trade my entire fortune-wortune for you to always stay by our side! Er, make that half my fortune...