This was a question from the definition stage of this proposal
Why is the Korean name 이 written as Lee in English?
The name is written in Hanja as 李 which is pronounced as Lǐ in Chinese.
Wikipedia has some information on why the spelling Lee is so common
Though the official Revised Romanization spelling of this surname is I, South Korea's National Institute of the Korean Language noted in 2001 that one-letter surnames were quite rare in English and other foreign languages and could cause difficulties when traveling abroad. However, the NIKL still hoped to promote systemic transcriptions for use in passports, and thus recommended that people who bore this surname should spell it Yi in the Roman alphabet.1
The overwhelming majority of South Koreans with this surname ignored this recommendation and continue to spell it as Lee. In a study based on 2007 application data for South Korean passports, it was found that 98.5% of people with this surname spelled it in Latin letters as Lee in their passports, while only 1.0% spelled it Yi.
A few people with this surname historically spelled it Ye, as in Ye Wanyong of the Korean Empire. Rhee has also been used as the Latin letters as in Syngman Rhee and Simon Hang-bock Rhee
The traditional pronunciation of 이, I, or EE is still employed in South Korea. It does not sound the same as the "Lee" (as Lee Myung-bak). The reason "Lee" was adopted instead of "Yi" is because of the English surname ("Lee") is more familiar to non-Koreans. Furthermore, non-Koreans pronounce "Yi" as "Yai" when reading the name, and often mistake Koreans as Chinese from the name when traveling or living abroad. Therefore, the majority of Koreans use "Lee" instead of "Yi".
It doesn't really answer why this name is more commonly Koreanised as 이 instead of the more logical 리 (which can be pronounced as either Lee or Ree). May be that should be another question.
There is '두음법칙 (First Initial Sound Rule / Law)' in Korean:
<언어> 일부 소리가 단어의 첫머리에 발음되는 것을 꺼려 다른 소리로 발음되는 일. ‘ㅣ, ㅑ, ㅕ, ㅛ, ㅠ’ 앞에서의 ‘ㄹ’과 ‘ㄴ’이 ‘ㅇ’이 되고, ‘ㅏ, ㅓ, ㅗ, ㅜ, ㅡ, ㅐ, ㅔ, ㅚ’ 앞의 ‘ㄹ’은 ‘ㄴ’으로 변하는 것 따위이다. [비슷한 말] 머리소리 법칙.
부가정보 : 한글 맞춤법 제5 절 제10 항에서 제12 항에 따르면 “한자음 ‘녀, 뇨, 뉴, 니’가 단어 첫머리에 올 적에는 두음 법칙에 따라 ‘여, 요, 유, 이’로 적고, ‘랴, 려, 례, 료, 류, 리’가 단어의 첫머리에 올 적에는 ‘야, 여, 예, 요, 유, 이’로 적으며, ‘라, 래, 로, 뢰, 루, 르’가 단어의 첫머리에 올 적에는 ‘나, 내, 노, 뇌, 누, 느’로 적는다. 예를 들어 ‘여자(女子), 연세(年歲), 요소(尿素), 유대(紐帶), 이토(泥土), 익명(匿名)’은 ‘녀자, 년세, 뇨소, 뉴대, 니토, 닉명’이 아닌 ‘여자, 연세, 요소, 유대, 이토, 익명’으로 적는 따위이다.
When consonants such as 'ㄹ' and 'ㄴ are placed at the first block of a word as an initial consonant, it is pronounced in a different way. For example:
In front of 'ㅣ, ㅑ, ㅕ, ㅛ, ㅠ' , 'ㄴ' and 'ㄹ' are pronounced as 'ㅇ': 녀자 -> 여자, 려권 -> 여권, 리소룡 (Bruce Lee) -> 이소룡
In front of 'ㅏ, ㅓ, ㅗ, ㅜ, ㅡ, ㅐ, ㅔ, ㅚ' , 'ㄹ' is pronounced as 'ㄴ': 락원 -> 낙원, 로동 -> 노동
*This rule applies to only Chinese characters and doesn't apply to original Korean words such as '리을 (ㄹ)' and '녀석', etc. Sometimes, it applies to a compound noun such as '신여성', '남존여비', '남녀노소' and '중노동', etc.
In short, if '리' is placed at the beginning of a word, it is supposed to be pronounced as '이'. That's why the family name '리' is written and pronounced as '이'.
Bruce Lee (李振藩) has the same Chinese character family name. It is written as '李' and pronounced as '[Li]' in Chinese because there is no 두음법칙 in Chinese. Korean writes the family name in the same way as '李', but it is pronounced as '[이]' because of 두음법칙.
One thing that you should note is all family names are proper nouns. You can use whatever English spelling you want for your family name. For example, the most famous Major League baseball player among Koreans '박찬호' writes his family name as 'Park' and the most famous LPGA player '박세리' writes her name as 'Pak'. Park has been favored by many Koreans as it sounds familiar, but its pronunciation is not exactly same as '박' and some chose to use 'Pak' instead, but the problem is English speaking people pronounce it as '팩'. It is not easy to find a proper spelling for Korean family names.
Some Koreans use 'Rhee' or 'Yi' for the family name '이'. That's what they prefer. There is no hard-and-fast rule on this issue.
I always considered it an 'English distortion effect' when trying to nail down the sound from Korean to English as close as possible.
Words in English that don't have any consonants look weird (이 -> Ee?), while Yi or Lee provide a very similar sound, but distort it a bit to make the word more readable.
When I was teaching ESL in Korea, I asked one of my adult students about the last name 이, pronounced like the letter ' e' and why did they change the pronunciation and spelling to Lee. He offered this answer: If we told people that our family name was 'E' it would be hard to write in English without confusion and if it was a man he would be 'Mr.E' and it would sound like 'Mystery' when a man introduced him self. I am not sure if that is the real reason, but it the best and simplest answer and it does make sense!
1) Note that there are Ye, Yi, Lee, Rhee etc for family name 이. But 98 percent use Lee (cf. user17915's answer)
China : 리, Korean-writing : 리 and English : Lee is changed into
China : 리 and Korean-writing : 이
At that time, we do not change English writing personally, because we do not use English name frequently.
2) English-writing rule of Korean character is changed frequently : For instance, Busan and Pusan for 부산 city is changed at each period 10 years roughly.
@ In my thought,
(1) people do not know the pronounce about Ye, Yi, and Rhee
(2) I am different from you : Lee is more beautiful character than Yi, in Baroco style.
For, family name for China 금 and Korean 김 has two choices : Kim and Geum.
Kim is more popular, but when I encounter Geum writing, I can not forget him in long time (For 서, there are two : Seo and Suh)
(3) Pronounce : If someone works in abroad, then he may use one, which can be easily pronounced by others : Typically, Kim Yuna. This is 4-th in Korean users.