Hi, ledojaeger,
One thing I notice about the translation of "ganbaru" is that people tend to translate it as "I'll do my best" (or "do your best") in every situation. It's like the saying, "If all you have is a hammer, every problem looks like a nail." I'm not saying that "ganbaru" doesn't mean "...do my/your/our best," but I am saying that ganbaru is a very context-heavy word when it comes to translating.
In this case:
いやいや、頑張って自分でお金を稼いでみせるよ
I think "ganbaru" implies that the person is emphasizing that they will do it on their own (under their own power/working hard), without outside help or influence. For example, if you took "ganbaru" entirely out of the sentence, you would only change the meaning by degrees, not substance (i.e. the sentence would almost translate exactly the same with or without ganbaru).
So, assuming (a big assumption, admittedly) that there is more context here (e.g. "Are you going to ask for their help?" "No, no...I'm going to..."), you could even translate this sentence as "...on my own, without their help." No reference to "do my best" at all.
I think the "shite miseru" ending backs up my interpretation because it's an "intent" that implies a bit of defiance on the part of the speaker.
Maybe a case of TMI (too much information), but these nuances are why I find languages interesting.