Your UG needs to have an official corporate name. You can choose a name inspired by somebody's name, your corporate purpose or you can get creative and make something up. In every case the name suffix “UG” (haftungsbeschränkt) is mandatory, so your chosen corporate form is transparent to the public.
The four options are explored in greater detail below:
Name your company after a person
It is common to use the names of one or more co-founders as a company name, e.g., "Franklin Smith UG."
Create a fantasy name for your company
You may invent a word to serve as your company name, e.g., "Bripotaz UG (haftungsbeschränkt)".
Describe what your company does
Use a description of the purpose or the activity of your company, e.g. "Steve Baker's Online Children's Book UG" if you are selling children's books online.
Use a combination of the above options
You can combine two or more options, e.g. "Bripotaz: Frank Smith’s Online Book Store".
Before you incorporate a UG, you should choose your company name carefully. Something to consider is the fact that many founders use fantasy terms that are not only unique - and thereby avoid copyright issues - but also pronounceable in as many languages as possible.
Most importantly, it's crucial that your company name only has positive connotations. Always consider your future customers before you make a final decision.