Price is determined by supply AND demand. Just being rare by itself doesn't mean anything.

For example...modern coins are very very uniform. Occasionally, there will be a mint error, and defective coins get into circulation. Those are rare and in demand, so they are worth some money. On the other hand, if you look at ancient coins, the dies were hand-carved, and variation is the norm. So a "defective" coin doesn't mean anything there.

You can weld up your own bike frames and make all kinds of one-of-a-kind bikes, but can't hardly give them away, then.

If you have some old bike that was the first of its kind or a desirable brand, it may be worth something. If it's just some department-store label made by makers unknown, not so likely.