I am a:
- non-EU student
- currently have a work permit for a part-time job
- due to finish my studies in the next 6 months

I may have an opportunity to take on some contract work in the next 2 months, but am not sure what the process would be to legally work and get all the permits.

The background for this contract work is me having an idea during my studies for a tool. A big financial services company is interested in me developing this tool for them.

What sort of permit will I / they have to apply for? Moreover, what are the chances of me getting such a work permit?

None I'd say. Your student permit allows you to work a max of 15 hours per week. My guess is that this new employer would have to apply to hire you and you'd have to give up your current job and do the 15 hours a week for the new one.

Indeed, student permit already covers 15 hours per week.