I helped build a website for a department when I was in college. I was paid a reasonable hourly rate in exchange for my services. In this case, the OP would be paying the university money in exchange for the privilege of building them a website. That's insane.
In general any time you do something for CS it's either trivial or if you had some qualification to back you up you could get paid; you're not paying for the privilege of designing their website you're paying for the support, guidance and ultimately assessment of your skills.
There is a problem of larger scope which is that education at a school level has been poor or non-existent as side effect of this is many people self-teach and then start university with skills way beyond their peers, web design is a good example of this because it's a relatively open platform that's easy and fun to get into, the result is that most universities must start from no assumed knowledge in the field.
It's only because you were able to teach yourself webdesign that you feel that the task is beneath you. If you really want to address this problem you need education to be better across the field so that university level can address significant tasks rather than filling in the basics for the first 2-3 years. This is starting to be addressed in the UK by the likes of the Raspberry Pi foundation and at a government level by mandated CS education in primary school but without incentives for teachers to take up the subject I don't know how effective it will be.