Remove the display panel to access the inverter (it's behind the screen).  Then source a part using the part number on the inverter.

Presumably you've already removed it if you know it's faulty - if not, and the symptom is a black screen, then it's more likely the bulb than the inverter on a system this old.

Service manual is here:
http://support.dell.com/support/edocs/systems/ins6000/EN/sm/index.htm

Thanks.

Im not 100% its the inverter. The symptoms I see are that the screen turns (almost completely) black after about 20 seconds. If I shut down and power up again, it works for another 20 seconds. It now seems I can run longer if I run on the battery (on lowest brightness) instead of the power cord. There is also a weak high frequency whizzling noise from the laptop before it turns black. I did a battery replacement about a month ago. In the last month I have also seen that the display shows a red tone in certain areas of the screen. The extent of this seems to vary depending on how long the laptop is turned on.

Inverter issue?

Sounds more like the bulb - the red or red-orange hue is usually a clue-in that the bulb is dying.

This isn't a do-it-yourself job unless you've got the tools and expertise though - leave it to a pro.

Which would likely cost me as much as a new laptop would cost. I dont like the idea of throwing away a laptop every x years due to minor issues like this one.

If you can find a new notebook for the $100-150 a backlight replacement costs, more power to you.  You probably can find a used replacement screen for that - but it'll come with an unknown number of hours on the backlight, and there are other pitfalls as well - starting with compatibility.

Hard drives generally last 3-5 years, so if yours is original, it may be near the end.  Otherwise, yes, other parts will fail - batteries, mainboards - with time.  Any notebook over 3 years old will see more failures than those under -- and even within three years, about 20-30% of notebooks will see a major component failure.

Ok, I got an offer from the local Dell (Germany), and the repair would cost me 365 Euro (all incl., i.e. pick-up, V.A.T etc). This is in the order of magnitude what I expected, and far more expensive than the 100-150 USD.

Contact a shop that specializes in LCD repairs - in the US, the cost runs $100-150, as I stated before -- what Dell is quoting you is the price for a new, replacement screen -- not a repair.