In the world of coffee, my compatriots spend princely sums on their water systems. Luckily, Baltimore's water supply is near perfect for coffee, and my resultant comparative spend has been quite minimal over the years.
First off, how does the water taste from your tap? Does your family truly hate it? Or do they find it not appealing because it doesn't taste like the bottled water they've grown accustomed to? I know it seems weird but different bottled waters taste differently (to me they can taste wildly different - like the difference between some crappy no-name router and the OF 2200).
Reverse Osmosis systems basically strip the minerals out of your water and then you use a mineral formulation that you add back to your water supply to craft the flavor of your water. It's cool but it's both expensive (at least good RO systems are) and wasteful - because the system will dump a not-so-small-percentage of the water as waste. Mainly it's because the filtration system isn't able to keep up with the water supply flow. Imagine a Brita pitcher - open it up, put it under the faucet and let 'er rip. The flow from the supply (faucet) will quickly overwhelm the Brita and the excess water will overflow its reservoir. This is, essentially, what happens in an RO system.
I have a system in my home because we're on well water and it's super hard - like the kind of hard that when I run out of salts, the soap refuses to become sudsy. I've tried to accustom myself to drinking it, but...
What you might try doing is surreptitiously taking some empty Sam's bottles and filling them with your tap water - and passing it to your wife and daughter. See if they even notice a difference. Don't say anything. Just let them drink it and see if they comment about it. I do think that most people who say the bottled water is "better" think that because they've convinced themselves of that fact. Coming from an era when we drank for garden hoses and park fountains, I'm aghast at how many bottles we waste. That said, I do bring in Deer Park 5ga bottles for home drinking.