I've been seeing a lot of Esperantized names, where people adjust their names to Esperanto spelling and even change it more to make it "Esperantized". In many ways, this is great, because names can be very different and it's important to get it right. What's the origin of this tradition?
I don't think it's much of a tradition, but rather convenience. Unless you transcribe it, i.e. write it phonetically (as it should be read) as the closest approximation using characters Esperanto has to offer, you won't really know how to read it. This is because names are just words in other languages: you can't pronounce them unless you are familiar with the language.
Transcribing names is also done in many other languages, so it's not an unknown thing to do. As a native Serbian, I can confirm we always transcribe names; I've also seen it in Russian and Greek. Another (more extreme and more obvious) example is Japanese, where we also transcribe (romanize) characters.
For example, take the name Michael. You can't be sure how to pronounce it unless you know which language the name is taken from (could be English, Italian, French, German and probably many more) -- it's written the same in all those languages, but it follows different pronunciation rules. So it could be Majkl, Miŝel, etc.
I'm not aware of any existing strict rules for transcribing names, so I usually go "by ear".
Some people like to add the final o to make it look like every other noun in Esperanto. Personally, I don't like using it because then it loses the original pronunciation of the name; also female names tend to look ugly or masculine (like Maria becoming either Mariao or Mario). I usually add o only when accusative seems awkward or impossible (John would be Ĝon and Ĝonon in accusative).