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I learned, in 35+ years in copywriting and marketing communications, that if you invent a good headline, almost anything will impress potential or actual employers. One of the best ways to write a good headline is to find a pun or word-play that fits the organization's product or service. Here's an example: When I was working at Mattel Toys I was asked to come up with copy for our annual catalog. Driving home from work one afternoon, I mulled over every expression I could around the idea of toys. Suddenly, it came to me: "Our work is child's play!" It was a huge sensation. The words I wrote for the inside, basic spiel of the catalog were pretty mundane, but my bosses were ecstatic about the headline.
A few years later I was a copywriter at a Silicon Valley company--Tandem Computers, which was always trying to impress the world with its "fault-tolerant" systems, which, supposedly, never went down. So they were used in stock markets, ATM's, etc. I needed to come up with some ad copy that was non-technical, because the audience didn't know or care about hardware redundancies and all that stuff. My solution: "Computing at the speed of life." Just a tiny twist on a common expression.
So...spend some time just brainstorming with yourself, starting from whatever your company is trying to sell, and just try out every pun or word association you can think of and write it down. If you actually write it, the act itself somehow helps spawn more variations. Believe me, a clever headline can overshadow an ocean of boring, mundane text. Good luck!
There are a lot of companies hiring for this type of role - a virtual work-from-home writing job. Content Writer would be the title. Copy Writer is more of an advanced skillset with persuasive sales & marketing type language. You could also contact a business and offer to write something for them. For example, a local small business in your area might need some marketing help. Then you would have before and after examples and/or a portfolio of work.
I think possibly writing articles and submitting them to online publications that accept contributing writers would be a great fit for you! Sites like Buzzfeed and RYOT allow you to write about topics you enjoy while also beefing up your portfolio. Also, if you're still in school, writing for your student newspaper (if there is one) is a great idea. It added SO MUCH to my portfolio. Also, you can look for volunteer opportunities to write for corporations or non-profits. Not getting paid is a downfall but getting the experience will help you in the long run.
Hope this helps!
Perhaps you could contact volunteer groups in your area (or national ones) and offer to write content for them. I would be impressed with someone who had written for groups such as The Ronald McDonald House, Habitat for Humanity, or the Girl Scouts.
Another way is to contact small employers nearby and ask if you could compose website or ad content for them. Often times a small business owner has entrepreneurial skills, but lacks writing talent - or just doesn't have the time for it. Go online, look at their website to study the "voice" of the company and then send them a proposal.
In addition, there are many freelance listings for writers. While the pay may not be lucrative, it will help you flesh out your portfolio.
I have been able to get bylines from teaching and writing entities (my forte) that have an online presence. They are always looking for content. I send them an email pitching my idea with a link to other work. This has been successful for me.
Best of luck!