Yes. In theory a person can not only be arrested, but convicted at trial (proven guilty beyond a reasonable doubt) based on one person's testimony. Hard to believe, but true. Practically speaking, it is rare that an arrest is made or case is filed in court with just one person's testimony and no other evidence. Additional evidence could have been injuries, other witnesses' testimony...etc. so my first thought is are you sure there was nothing else other than the girlfriends' statement?
I'm not clear exactly what the last part of your statement says - are you saying the charges were dismissed because of insufficient evidence? If so, that is not surprising. Perhaps they arrested him, and tried to get some sort of statement or confession from him, they didn't, and then decided that they could never convince a jury beyond a reasonable doubt that he committed a crime with what they had. Or are you saying the case was refiled because there was additional evidence?
Yes, and it happens all of the time. Set aside the situation you are considering - the rape case. Imagine that you are in a parking lot, going to your car, and it is late at night and no one is around. There are no cameras or recorders. Out pops a guy from behind a car & says give me your money. Wisely, you give him your money. He runs off. You call the police. You would not be too happy if the police refused charges because there is no physical evidence or other witnesses, right?
Most rape laws permit the reporting of the incident LONG after the incident occurred. In those situations, there is certainly no physical evidence yet they are prosecuted just the same.
The law says that an accused can be convicted on the testimony of another person, without ANY additional evidence, if the fact finder believes that person beyond a reasonable doubt (or by whatever definition the particular state uses.) And, the appellate courts will uphold convictions based on the testimony of a single witness.
I'm unclear on whether there were charges that were dismissed and now have been refiled, or if they are just dismissed. If they were refiled, this man definitely needs a good lawyer. If they were dismissed, he should consult with a lawyer about the possibility of removing this accusation from his record. It may or may not be able to be done.
Although I have answered the question to try to help you, you should consult with a lawyer in your area in person on the matter. In addition, my answer does not establish an attorney-client relationship between us.
My colleagues are correct but a case shuch as you describe is very weak. If there is no physical evidence to support the accusation then it is her word against his. Most prosecutors would not take that to trial.