I am going to guess that because Panther is so old there are no longer any updates available for it and it may even be the Software Update interface doesn't work any longer. With updates you're better off searching on a web search engine and going to Apple and directly downloading them.
Apple stopped developing pretty much anything that would work with Panther at all 6 years ago, and probably absolutely nothing 5 years ago?
The highest version of Safari you can run under Panther 10.3.9 is Safari 1.3.2---pretty old considering the current version is something like 5.1.7.
You can run a browser that, although not necessary current, has been updated since Safari 1.3.2:
This version of Camino runs under Panther: Camino 1.6.11
iCab 4.9 also runs under Panther: http://icab.de/dl.php
If you post the PowerBook's screen size (12, 15, or 17-inch), the processor speed, and amount of RAM you have installed, we can see if you have something worth updating and how to do it.
would you recomened download the next O/S after panther?
No. There is no legal download of any Mac OS version higher than 7.5. Downloads that are offered are illegal and may damage your computer. The next version, OS 10.4 "Tiger" is in high demand and a proper full-retail install disk is not longer available from Apple and can cost more than some used PowerBooks are worth today.
You can add RAM to reach a total of 1.25 G RAM; 256MB is barely adequate for Panther.
You can use up to Mac OS 10.5.8 "Leopard" although it may be a taxing even if you maximized RAM. The 60GB hard drive is probably OK as long as you can keep about 15-20 percent free space.
Your video hardware (NVIDEA GeForce FX Go 5200) has 64MB VRAM but lacks some of the newer video technologies that later versions of OSX use. That would not keep the computer from working but you'd notice some choppiness with some actions.