Early Mac Minis were all PowerPC G4 processors. This meant, they will always have trouble loading the latest websites, as there is no Silverlight for PowerPC, and you probably need to look at these solutions for Adobe Flash:
https://discussions.apple.com/thread/4781929
Go to Apple menu -> About This Mac to find out if you have a G4 or an intel processor. Java on 10.5.8 is very limited https://discussions.apple.com/docs/DOC-6491
If you have an Intel, you at least can upgrade to 10.6.8, which is still limited in terms of Java, but a little better:
https://discussions.apple.com/docs/DOC-5532
If you have at least an Intel Core2Duo or Core i3, i5, or i7, you can upgrade to Mac OS X 10.7.3 or later, but only after checking your software compatibility with this tip:
https://discussions.apple.com/docs/DOC-6271
10.7.3 offers both the latest Java and Flash support to date, and thus will not have trouble browsing any website except the few that require Windows, which can be added through this tip: