This project that you are working on from your thesis, I assume you do not own the rights to it and are developing it for the company?
I’d look into the value of that. As it is your thesis, it was originally yours, I assume. Did you sell it to the company?
So now, I guess, the company has decided to proceed and to give you the rains while at the same time also asking you to help out here and there on other projects.
Is this correct? Or will the “helping out” be your main occupation?
I have just accepted a “trainee” position as well which I will start in the fall.
As it is in the US and I am European, the term “trainee” or “intern” is much more for the visa application than an indication of responsibilities and compensation. So I am being paid a full wage.
Point being, don’t get hung up on the term “trainee” as it really is much more of a legal wording to classify your status within the company, since you are not being hired as a fulltime Junior Designer and the commitment is only 6 months.
Without knowing you, the company and the projects it really is impossible to say how much you should ask for and what you will be offered. So take what is written here with a grain of salt.
As stated earlier, it seems like you are getting a lot out of the arrangement other than financial compensation and it is for you to figure out where your bottom line is.
All this being said, getting a job offer straight out of school is a success in itself and just getting that first step onto your resume is quite valuable.