"think up a name" is a grammatically correct phrase and can be used in written English.
It means to come up with or create a name. Example: Can you think up a name for our new pet cat?.
Only when the boys received their £25 cheque did they realise they would need to share a bank account, and think up a name to put on it.
Ford even wrote to the poet Marianne Moore, asking her to think up a name that would "convey, through association or other conjuration, some visceral feeling of elegance, fleetness, advanced features and design".
The Higgs boson scores well on 1 and 2, but in my view fails miserably on 3. Equally, God particle fails spectacularly on 1 and 2, but does rather better on 3. I'm off to think up a name now.
Now the International Union of Pure and Applied Chemistry, which decides on such important things as names and symbols, has officially recognised the discovery and has sent Sigurd Hofmann, the lead researcher of the team that made number 112, a formal letter asking him to think up a name for his new element.
Think up a name for your world.
When registering your business name, think up a good business name that is not common.
Unlike the fleeting couplings of convenience that over-populate the pop arena, these guys clearly spend time together; they even thought up a name, rather than "Danger Mouse feat James Mercer".
Once you have your domain name, think up an equally ridiculous Web site description.
Therefore, I thought up a clever name for a contraption I invented that locates my "To Do" list around the house with an annoying buzzing sound.
The first step was to think up a memorable name.
