1. Actually legally changing your name.

2. Adding a legal alias which you can use for official business.

This issue is important to many foreigners in Japan because a name can mean a lot.  Many Koreans and some Chinese people want to change their name to a Japanese looking/sounding name in order to blend in better and avoid racism.  Even if you are obviously foreign, the ability to use a Japanese name will make some things easier.  For example, some Japanese web sites require you to enter your name in Kanji, and Romaji isn't accepted.  Also, if you have an unofficial Japanese name and you want to use it,banks and other official institutions won't let you unless you prove that it's legal.  

Further, it's sad but true than many financial institutions decline to offer credit to foreign people, under the suspicion that they could leave at any time.  If you make it less obvious that you're foreign to whoever is dealing with the paperwork, you are more likely to slide past this filter, while still doing nothing illegal.  Even if they do realize your are foreign, they will probably also realize that you aren't a clueless foreigner, but instead actually know what's going on and are in Japan for the long haul.

Middle names are common in many countries, but not normal in Japan.  As a result, there is no infrastructure set up to deal with them.  Yet, many places insist on using your passport name, so they tack your middle name onto your first or last name, and then it doesn't match your name as recorded somewhere else, which can lead to problems.  

One common problem with using katakana and romaji for foreign names is that people are never sure of the order.  Japanese people put their family name first in all normal correspondence, business cards, etc., so that's what you should do when dealing with others in Japanese even if you're foreign.  On the other hand, some foreigners misguidedly put their "first name" first, and even when they don't, many Japanese assume they did, and reverse the order again.  This leads to odd things, like Japanese people referring to you as "Mike-san", "Tom-san", etc.  At the least, it's harmless, but odd.  At worst, it's unprofessional and might cause confusion later.

Another issue that occurs is "rounding off" of romaji.  Let's say your name is "Mike Smith".  This may be recorded on your health insurance card, etc.  Then, it gets converted into "マイク　スミース”.  Japanese companies (and people) use Katakana to determine pronunciation, since sometimes you can't guess from the Kanji that are normally used.  As a result, when they want romaji for something, the katakana form of your name is a reliable way to generate this.  Except, of course, that the sounds from the spelling in your native language were rounded off for conversion into katakana (and the Japanese sound system).  Then they convert your katakana back into romaji, it will end up as "maiku sumiisu".  Now, even if you didn't particularly want a Japanese name, you have at least three spellings in common use.  The original English, the Katakana, and the romaji version.  One place this often ends up is on credit cards, and anywhere else where there is a system set up to automatically convert kana into romaji for display.  Then this may not match with your "real" name, which again can cause issues in the long run.  One easy solution is to change your name (or add a new one) legally to match the romaji spelling.

1. Changing your name

For Foreign Nationals

Japan takes the stance that they aren't in charge of changing your name if you aren't Japanese, that's supposed to be managed by your native country.  If you are a foreign national and want to change your name, then you have to do it in your home country, and get a new passport issued in your new name.  This obviously rules out Kanji in most cases.  This is a drastic step to take, but it could be useful if you plan to stay in Japan for a long time, and want to solve the middle name and round-trip-romaji problems listed above.  Once you change your name and get a new passport issued, you can take it down to the government office and get new documents issued using that name.

For Japanese Nationals

Japanese can change their legal name, which will be stored on their official records.  This is usually done for marriage, but can be done for other purposes.  Their maiden name will be stored.

2. Adding a Legal Alias

Adding a legal alias mean that your retain your original name, but add a new one that can be used for official purposes: i.e. opening accounts, conducing business, etc.  However, since it's not terribly common, you may have to point out the law to some places that might not initially want to comply with your wishes.  At any rate, this is relatively easy way to establish your Japanese name as official and side-step some of the issues mentioned above, without the drastic step of changing the name on your passport.

For Foreign Nationals

You can register your alias at the government ward office.  In order to do this, you have to prove you are using it for official business.  This may seem to be a catch 22, since many places won't let you use a name that doesn't match your ID.  The simplest way is to mark your mailbox with your new name, and then sign up for some service that will send you something in the mail using that name.  Once this is done, you can take your "evidence" with you to the ward office, and ask them to note your legal alias.  If you have a foreigner registration card, they will make the note there, and in their files.  Once this is done, you will also be allowed to make a new Inkan (stamp) using the name you just registered, and register that with them as well.  Here is a the guidance from the Minato-ku government (in Japanese, of course) on how to do this.

Note that if you use Kanji, the kanji that can be used in names that registered are restricted to those in two lists released by the government on a yearly basis.  You should check these if your proposed name is not a common Japanese name or contains any unusual characters.  Even if the characters are on the list, they won't let you register anything they find offensive. (f.e. the Kanji for "Satan" has been rejected).

For Japanese Nationals

While actors and such often use aliases, there is no official legal way to do this that we have been able to find.  If you are a Japanese passport holder, you can change your name as mentioned above.