I often help my friends and family members open email accounts. Sometimes they are starting to use the Internet and sometimes they move from old email services (Yahoo, Walla!, ISP) to something modern (like it or not, Gmail).
At some point they have to fill their name, which will appear in the “from” field. And then I have to suggest them to write it in Latin characters, even though most of them speak languages that aren’t written in Latin characters – mostly Hebrew and Russian. Chances are that some day they will send an email to somebody who cannot read Russian or Hebrew, and Latin is relatively better known.
Only relatively, though. It may seem obvious to you that everybody knows the Latin script, but in fact, a lot of people are not comfortable with it at all. There are also other complications: lossy and inconsistent transliteration rules (is Amir אמיר or עמיר?), potential right-to-left rendering problems, and more. And of course, all people are happy to see their name in their language.
And people are also happy to see their friends’ names in their own language and not in a foreign or a neutral language. I have, for example, a lot of friends in India. Most of them write their names in English, but some write it in Marathi or in Malayalam. It’s certainly good for them, but in practice it’s much harder for me to find them this way, so English would be better – but Hebrew or Russian would be better yet.
Finally, there are a lot of people in the world who have more than one linguistic background. Mine are Russian, Hebrew and English, and I am really not such a special case. There are many millions of immigrants who have mixed backgrounds: Punjabi-Hindi-Urdu-English, Kurdish-Turkish-German, Kazakh-Russian-Norwegian, and others, and others and others. From each of these backgrounds they have friends, co-workers and family members, with whom they would love to communicate in the respective language. In each of these backgrounds they have friends who would want to find them using the name under which they know them there and using the appropriate language and writing system.
And sometimes people change their names, too. I did it (twice!), and so have many other people.
All this means that people’s names should be translatable, just like books, articles and software interfaces. Facebook and Google+ allow me to add a very limited number of names in foreign languages. Why wouldn’t they let me write my name in four, five, ten languages? This would make it easier for people who speak these languages to find me and to communicate with me. I would go even further and allow people who speak languages that I don’t know well to write my name as their hear it in their language and to add it to my details. Yet again, this would make me easier to find to even more people.
Some degree of automation can be possible. A lot of names are, after all, repetitive, so social networks would be able to suggest people with common names how their name would be written in other languages.
Wikipedia is actually quite good in this regard: Usually people have the same username across projects, and this username is not necessarily written in Latin letters, but people can customize the appearance of their signature in each project. I did it in a few languages, and people who speak those languages appreciate it.
I can only hope that social networks and email systems will allow as much flexibility as possible with this.
Aharoni in Unicode 
There is one major problem with your suggestion:
For security reasons, in official circumstances, and even on FaceBook, people are required to use one single recognizable name. If they want to change it, even just the spelling, they have to register the change, so it can be traced back. It is very hard to manage ten versions of a person’s name, each written in a different writing system. In Israel, you usually have two official versions of your name – in Hebrew characters and in Latin characters. I think that an Arabic-character version is also possible, but most people don’t use this option, because 80% of the population don’t read Arabic characters well. This state-of-affairs is not so easy to handle, and we’re talking about 2-3 versions only.
It is not a problem. It is a challenge, and not a verhy hard one.
The so-called security offered by “real names” policies on Facebook and Google+ is very dubious, both in principle and in practice. Pseudonyms still get through (like “Adolf Hitler”!), and real names are banned. I don’t think there’s a useful way for banks of rent-a-moderators in the Phillippines to figure out what is and isn’t a good name.
This has been driven more by the needs of advertisers, not security.
Better systems will require acknowledging that “trackable identity” and “name display” on the internet are two different things.
I was also wondering how you cope with input. Do you have two different keyboards (Latin and Hebrew) or what method are you using?
Yes. My operating system gives me English and Hebrew and I switch between them all the time. I also frequently type in Russian, and for this I use a transliteration add-on in Firefox.
It’s a completely obvious thing to me, but time after time people tell me that it’s hard for them. I guess I have to try to think of something to make it easier.