Warning: major spoilers follow.
Coin ~ money
There are a few ways in which the District 13 leader could be symbolised by the idea of money.
Power. Money can be used to buy power, or as a representation of power, and one of the most important things about this character is that she seeks power.
Lack of personality. Money has no use value; it's faceless (except in the literal sense, coins frequently having rulers' faces on them) and emotionless. Similarly, President Coin plays her cards close to her chest and we never really know what she's thinking or planning.
It's also interesting to consider her first name with this in mind. Alma, in Spanish, Italian, Portuguese, and related languages, means "soul" - so "Alma Coin" suggests "soul of money". Not a flattering epithet, and one which hints at her coldness and ruthlessness. Similarly, in Latin, Alma is a feminine form of almus, which can mean "nourishing" - so another interpretation is "nourished by money".
Two sides of the same coin
This is the whole idea behind the events that play out towards the very end of the series: Katniss kills Coin because she realises she's just as bad as Snow. What's a common metaphor for two people or situations being equally bad? They're two sides of the same coin. Literally the culmination of the entire series is tied up with the realisation that Coin is just the other side of the same coin.
As a hint towards Coin's nature, this only really works in retrospect - I don't think it would be reasonable for even the shrewdest of close readers to deduce that Coin was likely to end up as bad as Snow just from her name. Nevertheless, viewing it in retrospect, it's still quite striking as an indicative piece of wordplay.
Flipping a coin
Personally I don't hold with this theory, so I'm not going to be able to make a convincing argument for it. But since you've mentioned it in your question, I will at least quote what others have said about it:
Money will work for anyone. It doesn't care who uses it. It's just a tool. It merely exists to be used. Money's fickleness becomes especially significant with coins. We use coins as a representation of pure chance. Flipping a coin gives you fifty-fifty odds. (Just look at Two-Face of Batman fame.)
Coin has no perceptible allegiance -- she doesn't even seem that loyal to District 13, her home. She embraces Katniss when it was useful to her. When Katniss became a liability, Coin threw her to the proverbial wolves. When President Coin offers to flip for the chance to kill Snow, it's a subtle allusion to her capriciousness.
-- Cristina Hartmann, Quora
I haven't been able to find any direct quotes from Suzanne Collins about this, and I suspect that none exist. She's notoriously reluctant to give press interviews, and there's not much available in the public record by way of direct commentary from her about her books.
References:
