Need a cable to connect Mac mini 2008 to a monitor and I can't find one

What monitor do you have? If it is an Apple monitor, then see if you can get the exact model by entering the monitor's serial number on the check coverage page here:

Check Your Service and Support Coverage - Apple Support

Do you get any LED power when the display is powered on? It should be on solid, but it if the display's power LED is flashing, then it means the display has a hardware issue. Many Apple displays have the display cable hardwired so you cannot easily swap cables.

FYI, that is a DVI connector.....both on the display cable and the back of the Mac Mini.

There was no 2008 Mac mini.

According to MacTracker, the Mac mini (Mid 2007) was introduced in August 2007 and discontinued in March 2009. For a display connection, it had "DVI (VGA or S-Video with adapter).

The maximum resolution was 1920x1200 (DVI) or 1920x1080 (VGA).

It’s a DVI cable https://www.amazon.com/gp/product/B07CG5KZ82/

Sounds like the Mac Mini has a hardware issue (or the DVD to VGA Adapter, or video cable) assuming those older displays comply with the specs I listed previously. If you hold the Option key immediately after the startup chime and you see nothing appear on the display, then it eliminates any software possibly causing the problem. Even if there is no bootable volume, you should still see something on the screen.

Depending on whether the Mini is 2007 or 2009, certainly would

make significant difference; in both graphic card/port styles and

display style ports supported. Because 2007 was last of old type

of that era, & 2009 follows later trend, to 'mini display port' idea.

Examples of my two are of wider range difference older hardware.

Since Late 2005 Mac Mini 1.5GHz PPC G4; & Late 1012 Mac Mini

2.3GHz quad-core intel-based server, a leap even further further.

[Yet only one cable adapter allows both to use similar displays.

MacTracker helped but wasn't determinative, back in the day.]

Some older systems won't be able to handle the display resolution of the newer displays.

From EveryMac.com, the supported display resolutions of a 2007 Mac Mini are:

DVI: up to 1920 x 1200

VGA: as high aas 1920 x 1080

https://everymac.com/systems/apple/mac_mini/specs/mac-mini-core-2-duo-2.0-specs.html

I wonder if that "buzzing" you are hearing may be the hard drive. A severely failing hard drive can prevent video from displaying on screen.

You could try removing the internal drive and using a USB to SATA Adapter, drive dock, or enclosure to connect it to another computer to attempt to access the data assuming the hard drive has not failed.

Mac mini 2008 doesn’t exist. Clarify with more details please.

Time to dispose off that Mac. Drop it off at the nearest Apple Store, they will recycle for free.