Lot's of advice here, much of it speculation.
As you may notice from my user name, I am an amateur radio operator. I have had personal experience with RFI and am familiar with the laws (Yes, there are FCC laws about RFI).
The first step is to ID the source of the noise. The recommendation of using a portable AM radio is spot on.
What we are fortunate enough to know is that the RFI is due to the installation of one of the computer's components. While monitoring the RFI, disconnect the components one at a time until the noise goes away. Percentage wise your monitor is the most likely culprit.
Once the source is identified, the reduction or removal of the RFI can be dealt with.
If this is a NEW unit, there should be an FCC sticker on it somewhere. That is the Part 15 sticker which basically says that the device doesn't emit RFI. If it does, take it back the the store and tell them you want a device that is Part 15 compliant and that you have one that was certified as such that wasn't.
If the unit is used, it depends on the device. One trick radio operators used on CRT's was to take them apart and put foil inside the case, all bonded together. However, I don't recommend it unless you are familiar with the HV stuff in a CRT. You can make a nasty situation by getting the foil in the wrong place.
Making the radio immune to RFI is not as easy. First, you have to figure out the ingress path. Is is the AC cord, the antenna or just plain front end overload? Each path will have different approaches.
DO NOT modify the guy's radio. Any modification voids the UL listing for safety.
Adding a grounding conductor to the chassis may make things worse, or even unsafe. If the radio came with a two prong cord it was NOT designed to have the chassis earthed unless there is a specific earthing terminal accessible without taking the radio apart.
Using coax to feed the antenna is a waste of time. Coax is unbalanced feed line and the radio and presumably the antenna are both balanced. Coax will just add loss to the system. Feeding the antenna with balanced twin lead is the way to go in order to get the antenna away from the computer.
You say the antenna is a single wire. Are there two terminals and only one wire? If so, that is where the feed line needs to be connected so the antenna can be moved away from the computer.
The current antenna is too close to the computer.
If the ingress path is determined to be via antenna and adding proper feed line almost, but not completely removes the RFI, a current or voltage transformer may (or may not) take the rest out.
All that being said, I had one CRT that no matter what we did, you couldn't operate a radio within 25 feet of it. I tried everything and finally just gave up. Well, I didn't try painting the inside of the case with copper paint, but did make attempts at shielding and isolation that worked on other monitors.