A long time ago, probably when I was auditing botanical Latin, I recall someone saying what the Latin word for motorcycle would be. It was a long and literal description of what one is given that such machines didn't exist when Latin was "alive." Online searches aren't helping. Anyone familiar with this and care to share? Gratias tibi.
I will make a suggestion. Prīmum, ecce Pondus, composuit et pīnxit Frode Øverli:
Fābula dē Pondō quī birotam cōnstruit. – The story about Pondus who builds a bicycle.
Composuit et pīnxit: Frode Øverli. In latīnum convertit: Tor Ivar Østmoe. © Øverli 2007. MareSilva.
The word bicycle and bĭrŏta are almost literally the same. The prefix bi- stems from the Latin counting adverb bis, and note that:
The word bis (“twice”) drops the s when making compositions, like the Greek word δίς (dís, “dis”).
English cycle stems from Greek κύκλος: circle, ring. The Latin rota, -ae means wheel. Now we can get some inspiration from Whitaker’s Words for how to make it automatic:
autocinetum, autocineti N N 2 2 N [GTXDK]
autocineticus, autocinetica, autocineticum ADJ [GXXDK]
car-; of a car;
autoraeda, autoraedae N F 1 1 F [GTXDK]
The first, autocinētum, -ī, is from Greek αὐτοκίνητος, which I believe translates as self-propelling; the second, autocinēticus, -a, -um from αὐτοκινητικός, means self-putting-in-motion; the final, autoraeda, -ae is simply self-cart, as in self-acting or self-moving cart. Auto- is thus a useful prefix.
Hence my suggestion for motorcycle in Latin: autŏbĭrŏta, -ae, in short autŏrŏta, -ae.
Being in the territory of New Latin, any term can be of questionable authority. If an author doesn't like the terms used before and prefers a neologism instead, they can easily justify said neologism.
Some of the terms I'm suggesting are already mentioned in other answers. The commonness of any term could be on account of many different reasons, so keep them in mind.
Adumbratio, a site full of New Latin and timeless Latin terms, gives birota automata and birota automataria, as well as bicyclula motoria and autobirota. These variations are due to variations in authors, though if you go to the website, it does cite the sources.
The Vatican's Parvum Verborum Novatorum Lexicum does not give a translation for motorcycle, but it does give a translation for motoretta— which means motor scooter in Italian—as birotula automataria, where birotula is a diminutive of birota.
I personally find autobirota to be my favorite, as it is a single word and makes sense. That being said, birota automataria seems to be holding its ground as a considerable option.