You've got a few options here. I'll start with the "big bang" from my perspective and work down.
If you or one of your team owns a recent model android, test your mobile platform with it (preferably off-hours, unless management doesn't object to this), and record any problems you encounter in your normal issue tracking setup. Then repeat those tests on your official environment. The moment you find a problem with the new android system that doesn't exist on the official platform, you have your argument - although making it tactfully helps. Demonstrating the problem and that it doesn't exist on the official environment is one of the most powerful ways to show that you need the newer platform as well.
Another method involves a little research: finding out approximately what the most-used mobile environments are (I suspect exact numbers are a bit hard to find), and where your official environment fits into this. If you have the mobile equivalent of testing your platform in Netscape when the vast majority of users are running IE 7 or better (example pulled from a hat and may not be accurate), you have a problem which could potentially cost large amounts of time - and therefore money.
Basically, you're looking at convincing management that buying this device will be cheaper than fixing problems caused by not having access to the most recent hardware. The closer you can get to "X problems caught in-house will cover the cost of the purchase", the more likely you'll succeed.
And yes, as glowcoder says, convince us you need it as an exercise in getting everything you need to deal with lined up and nailed down.