I would recommend against including a verbal offer on your resume. Even aside from whether the professor remembers, an offer of a PhD position might not be very meaningful. Professors vary in how careful and responsible they are about making such offers, and I've known of cases in which faculty expressed interest to potential students in ways those students thought of as near-binding offers but which were not intended that way. (If someone says "Would you like to come do a PhD with me?", they may view the question as having implicit conditions such as "provided you meet the admissions requirements and I can come up with funding for you," while the student may not realize that. This question means the professor will try to make things work out, but it doesn't guarantee that they will try hard or succeed.)
Of course your offer may well have been far more serious, but it's hard to convey this on your resume without going into too much detail about it.
Tone down the claim to say that I had the best project and leave out the offer.
I would take this approach, assuming you have some official recognition such as an award for the best project in the class. (If it was at a prestigious university, then having such an award might mean more than the PhD offer.) If you don't, then there's insufficient basis for calling it the best project.
If you're applying to graduate school, why not ask the professor in question for a letter of recommendation? That would give him an opportunity to write about how impressive the project was, how it was the best in the class, and how he wishes you wanted to specialize in his area since you would make a wonderful PhD student. Hearing these sentiments from him would mean more than anything you could list in your resume.
Similarly, if you're applying for jobs, you could ask the professor whether you could list him as a reference.
If it wouldn't make sense to ask him for a letter or to serve as a reference, then it's probably not important enough to be worth worrying about indicating on your resume.