I want to upload some of my works to Thingiverse without making my real name public (displaying it on the profile page).
I think it is OK to register my name to the site unless the make it public, and it is required by the terms to provide truthful and accurate information.
I tried creating an account on the site, but I deleted it because I couldn't find the way to hide my name (set another one) from the profile page in a short time.
I see some accounts that doesn't seem showing their real name (for example, their name on profile equals to their account ID, or at least not in two parts: first and last name as required on registration), so I guess this is archivable.
How can I set my name for profile page on Thingiverse after registration and logging in?
You can put whatever name you want in when you make an account - You decide what your name is, people go by pseudonyms in 'real life' all the time, this is perfectly acceptable. There are actually very few places where you have a legal obligation to provide your name as it appears on your birth certificate. Opening a Thingiverse account is not one of them.
To answer the other part of your question, yes, it is possible to change the name displayed after you have created your account. I would be surprised if they bother to actually keep an archive of past names though.