I'd convince them it's an excellent hobby/learning experience.  Plus you can always make the financial argument.

3 5850's will use approximately 450-500W continuously while mining and your machine will add a bit more, but let's assume 500 for the sake of argument.  That's about 12 Kilowatt Hours/day * 30 days or 360 Kilowatt hours/month.  Multiply that by your KWH rate for electricity and you can guesstimate how much it'll cost.   So, at 10 cents a KwH you're looking at $36/month in electricity.

Those cards can pretty easily do 330-340 MH/sec, so let's say a flat 1 GH.  Using that against the current difficulty with $14 BTC prices (both of which are subject to change), you can expect to mine about 18 BTC or around $250 worth of BTC.  

Of course, you now have to deal with a loud/hot box in whatever room you stick it in and all the hassle of getting BTC sold and into your hands as cash but it's definitely worth it.