You can project on the following:
1. If you face any unknown bugs, then Oracle will not develop any new bug fixes.
2.If you upgrade to the latest version, you can adopt the new features available in the new version to improve the performance of the application and as well improve the performance of the DBA activities like backup....etc
3. Older version means, you are into more riscue.... as even if you have extended support license, Oracle support team provide you the existed patch fixes, if you face any unknown thing, either they'll provide you a workaround if exists otherwise again they suggest to upgrade.
I hope these points will help you, just eloborate these things.... how actually they'll impact the production.
Ask them why they don't drive ten year old cars. (Don't ask if they do happen to drive ten year old cars.)
The point is, not upgrading is a penny-wise, pound-foolish management decision that underestimates the risks and costs of lagging behind in normal maintenance. It becomes more foolish as time goes on.
Upgrading also has risks, costs and benefits. One of those risks is third party software providers making the same foolish mistake. But if you have foolish software vendors, that is another risk you should consider ameliorating.
Some parts of the 10g are a work in progress, and bugs, misfeatures, and lack of features are corrected in later versions. A major example would be plan management. For example, if you have different parameters to the same queries, the best plan for each may be quite different. The optimizer and how it can be told to fix plans are both much improved.
Some new features may be very useful, but the value of that is site dependent. I know some of the older stuff I've written would be much different now, and some is actually worth redoing.