My brother is a student in his second year of university, studying electrical engineering. For him, web development is a hobby which he excels at, so he thought of officially working remotely as a web developer. However, most jobs require previous experience (he had never worked for a company), but he has a lot of hobby projects under work. So, what would be a good excuse for him to be hired? I thought of showing them some projects of his own, or maybe work for no salary for a month?!
Your brother should consider internships and freelancing to get some professional experience in web development. Your brother stated that he preferred to work remotely, but internships are a great way to get experience and get a foot in the door at a specific company, which can translate to a job later on.
Freelancing isn't tied down to a specific company, but it does give the option to work remotely and you can choose to take on jobs when you want.
My suggestion would be for your brother to do internships with a company during off months in school and pick up freelancing jobs during the school year (if he has time).
It sounds like your brother already has a decent amount of experience given where he is currently in his schooling/career, but may need to learn how to showcase that experience. No one has formal job experience when looking for their first job, and it is an intimidating hurdle to get over no matter what the industry.
Web development, while highly technical, is also very much a creative process and getting work as a web developer requires showing off this creativity. He could start a blog, website, or even YouTube channel showing off his work and discussing how he tackles common problems with web development. He should contact the people for whom he did his previous projects and his current projects to see if they are open to him displaying his work more publicly (i.e. to his network of potential employers and coworkers); any future projects he takes on should have this stipulation from the beginning. He is building his portfolio and brand with these projects so it is important that he show these off, especially if he hopes to one day get paid to do this kind of work.
I 100% agree with @TheRealLester, internships and freelancing (which it sounds like he's already doing) are amazing ways to gain experience without asking a company to invest lots of money and energy into a new candidate. Companies who are interested in working with younger developers tend to be more open to interns and often times work with local universities or community colleges to find potential interns. Your brother should research internship programs at his university (even outside of his department) as well as at other universities and community colleges nearby your area. If he has his eye set on some specific companies, he should also visit their websites (and maybe poke around their LinkedIn pages) to see if they over any internship programs themselves. He could try sending an email to a program at a different school or at an interesting sounding company just to ask about their internship programs, and what the process of applying is. Even if he is not eligible for a specific program, it will give him an idea of what qualities companies are looking for in their interns.
Quick tips from what I've seen working as a SW (not web) developer and being involved in the hiring process for my company:
Discuss, maybe in a CV, one full project in a reasonable amount of depth from beginning to end, and explain why it benefited the customer.
Because it's web, include URLs that the recruiters/interviewers can access so they can look at his work before he is asked to come in for an interview. Effectively, show them the work instead of just telling them about the work.
Take classes, on campus or online through anywhere, that take his skills to the next level, and (especially because he is still a student) include these classes on the resume. It's valid to discuss the more interesting courses in a CV or during the interview.
Cast a wide net when looking for work of any kind. My parents often say that it is a full time job trying to get a full time job, and they aren't wrong. He should tune the resume/CV to each company, but apply to a large number of companies (even positions depending on the size of the company) so he has more chances of getting a call back. Be diligent and persistent on this; it's work, but once that job offer arrives it will be worth it.