You'll have to first look carefully at what are the requirements of a bachelor's thesis if you choose to embark on writing one. Doing original mathematics research that you can write up in a form that is basically suitable for publication is a complex process that usually cannot be undertaken solely by an undergraduate. You would need a good adviser to help you in your project. Such an adviser should ideally be able to (1) come up with one or more problems that are suitable for undergraduates (this often involves knowing the literature in one field of mathematics fairly well and translating a more general problem in the literature into a special case that is suitable for an undergraduate's current background level); (2) be able to offer hints and suggestions when your first attempts to solve the problem inevitably fail (i.e. you're supposed to struggle with a problem if it is one that is in need of an original solution); and (3) guide you through the process of turning your discovered results into a well-written publication.
At many universities, the requirement that an undergraduate contribute original research in this strict sense of solving an unsolved problem is typically relaxed. Often, it is acceptable for an undergraduate to write an expository paper on an advanced topic in mathematics. Thus, issues (1) and (2) above are replaced with an adviser being able to (1') select an appropriate level of advancement for the type of project you are interested (i.e. not too simple and not too difficult) and (2') suggest specific references to consult and specific examples to work out to help you understand the topic well enough to be able to explain to others.
The latter task is much more feasible as a first exposure to research. It involves many of the skills that you will use in future research if you decide to pursue a post-graduate degree in mathematics. But it relieves you of the pressure of coming up with new results. You would still learn a lot about mathematical writing and it is certainly not trivial to learn about an advanced topic well enough to be able to write a good expository paper on the topic.
If you decide to pursue a project along the former lines, then you should be sure that you pick an adviser who has some experience with guiding students in this type of project. (Because your university does not require such a bachelor's thesis, this may be a non-trivial task.) You should have a long period of time to work on the project and should get started as soon as possible. Without any experience in mathematical writing, you should aim to complete your research first and obtain results suitable to write up several months before the bachelor's thesis is due. You will need that additional time to learn how to write up your work using LaTeX and will likely have to create many drafts before you produce a suitable document. Trying to do original research at the same time as you are trying to learn how to write a substantial mathematical document for the first time is a recipe for likely disaster.