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I 'd like to know the definition of this idiom "give him a run for his money". Thanks.
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Give him hell, compete fiercely against him.
how so?
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Loosely:

We may not win, but we'll try really, really hard, and we'll make sure that it's not easy for him to win.
Hi,

I see the underlying mental image here as a running race, perhaps between you and me. Let's each bet $100, and try our hardest. I'll give you a (difficult) run for your money, to beat me you'll have to run very fast.

As mentioned earlier, it's a phrase relating to competition.

Best wishes, Clive
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Incho, Marius, Barbara and Clive,

THANKS.