Let me start with a little story or two.
When I was in Kansas City, I took a job with a security company to fill the last few months before I moved to Saint Louis. It required a license. When that class broke for lunch, the instructor asked me to wait outside before returning from lunch, then when everyone was settled, I was to run in, push him out of the way, grab his briefcase and run out. I did so, it was fun.
He then asked the students to describe what happened (immediately following the incident), and describe the assailant (me). At first, their descriptions were pretty vague, then someone realized I was missing, and the others caught on so the result was a fairly accurate description of me although the event varied. After class, I told him that I felt it would be better if he did this with someone the people in the class had not seen before.
He asked me if I would be willing to do that for his class next week, as an experiment. I agreed. We did the same scenario. The descriptions where awful, It turned out that I was a tall, short, fat, skinny guy, balding, wearing a hat, clean shaven with a beard, etc.

Funny thing was, this class was made up of both seasoned professionals there to renew their licences as well as people new to the industry. Investigators, Security Guards, Loss Prevention Professionals, Corporate Security Professionals (AKA Body Guards), etc.
People make great eye witnesses when they are on "guard". For example, you hear a weird noise outside and go look into it. You are on guard, you mind is hunting for details. As a result, when you see that guy prying open your neighbors window you tend to take in all the details (though, if you beat the guy down you can take some pictures while waiting for the ambulance to arrive

). However, we are creatures of habit, and when faced with an unanticipated even we are briefly shocked and confused. It takes a moment of two to focus on the events. Often times, the events are over in that moment, hence the problems with eyewitnesses.
Another problem is that many of us do not retain this information well. The solution to this is simple. write it down immediately. Write down time and date, where it happened, any identifying info (like license plates and the like) what you can describe about the persons involved, etc. This way you will have a much better chance of retaining this information, both because it is on paper and also because you re-enforced it in your head by writing it down. When I see something suspicous, that I can't follow up on, I write down the details I do know so if it does come up that something happened in that area later I can pass what I saw onto the police. It is a good habit to get into.
As I finished installing an alarm and was leaving a customers home a few weeks ago, we saw a lady collapse in the street. He ran to her aid, I called 911. When I got there we both realized she was terribly drunk. She was also clutching a handful of cash. The customer of mine took the cash and counted it in front of me ($341.00). I wrote that number down in maker on the back of my clip board, because if anything ever came up after he turned it over to the authorities I wanted to be able to honestly back him up and I knew I would never remember the exact number.
What value do eyewitnesses have in criminal investigations and trials?Depends on the eyewitness. There are of course possible issues with their credibility (like the eyewitness is known to hate the accused). Then it comes down to their credibility, Whether they were prepared or surprised by the incident, etc. Also, some professions are better at this then others. For example, a modeling agency manager is more likely to notice details about a person then a mechanic who is more likely to notice details about a car then an electrician who is more likely to notice details about how appliances and lighting are installed. Notice these details become second nature to some professions. That is why veteran police officers, who have to deal with this daily, tend to be better at it then the rest of us (not to mention their training).
Why do we instinctively trust evidence given by people more than that given by inanimate objects, despite the known weaknesses of eyewitness testimony?Well this is not always true. We trust video evidence more then eyewitnesses (though defense attorneys will work quite hard at spinning what is on the tape to try to convince you that you do not see what you think you see

).
Otherwise I agree with
Argonaut that we tend to relate to people instinctively, which causes us to trust them over carpet fibers and the like. Though these days people understand the concept of forensic science much more then they ever have (Thanks to CSI and the like) and have gained alot of trust for that as well (which can also be flawed due to poor collection techniques, bad analysis, etc).
What can we do to mitigate these weaknesses?The best solution when faced with something you might have to recall for the authoritieslater is to write it all down immediately. And be honest, if you didn't notice what the guy looked like because your eyes were focused on the huge barrel on the .44 Magnum then just describe the incident and the gun and admit that you are not sure what the assailant looked like. You
will be asked to pick them out of a photo or "in person" line up during the investigation at some point anyway, so if you can't pick them out of a group of 6 similar looking people then why bother suggesting you know what they look like in the first place?