Although it is a very personal choice whether to use your real name or not, there are specific reasons why I choose to always use my real name. I have been writing online for many years, and have always used my own name rather than a pen name. There are pros and cons for each.
Using a pen name gives one a sense of anonymity. One may feel that there are certain privacy concerns that can be avoided by using a fake name. This is true, but only partially. The fact is, anyone with enough computer savvy can find out who you are regardless of what name you use on any given site. Still, if the plan is to only write short, journal-like posts, without future expectations other than payment for these posts, then it may be best.
For those who have written for years, or plan to write extensively, especially if those articles or posts contain some expertise, then using your real name establishes authorship. There was a time, just a couple of years ago, when one could Google my name and there were literally pages of listings of my online work. Ah, those were the days! In any event, this is true for many of my fellow online writers. Using your real name would also come in handy should you ever decide to write a book.
A following, such as this site has, can literally follow you to other sites. For instance, those who wrote for Squidoo often followed each other to Hubpages, as well as other sites. The “friends” you make on one site usually get to know the style of your writing, as well as the types of topic about which you write.
There is no right or wrong answer here. It is a choice that should be made thoughtfully, and with consideration.