Learn English – Literal meaning of “to give a run for someone’s money”

I actually think the "figurative meaning" you cite isn't quite correct. I would say that the (most common) figurative meaning of "to give someone a run for their money" is "to challenge someone."

But the literal origin of the phrase comes from horse racing. To want a run for your money is to want a horse that you have placed a bet on to participate in the race. Sometimes a horse is withdrawn from a race after bettors have already placed money on it; those bettors did not get a run for their money. From the OED:

2007 Racing Post (Nexis) 14 Jan. 9 Jayo was sent off the well-backed favourite in the 2m juvenile hurdle, but supporters never got a run for their money as he was pulled up lame behind.

Conversely to give someone a run for their money is to give a good race (even if you don't win) in return for their backing. This latter usage also suggests challenging the other horses in the race; hence its contemporary figurative meaning:

2009 Herald-Times (Bloomington, Indiana) 15 May d4/4 A home-wrecking catfight to give Uma Thurman and Daryl Hannah a run for their money.