In 1898, by imperial proclamation, ordinary Japanese people first took surnames. Millions of people chose new names based on where they lived, the recommendations of priests, history, Today Japan has a rich variety of beautiful surnames.
China already has last names - but they're mostly single character, and just a few of them make up a huge chunk of the names in the population. The top 20 names make up about 85% of the population - so everywhere you go you meet people with the same name. Some towns have had complete surname depletion - the entire town has the same family name.
Meeting someone with the same family name doesn't mean anything.
So my proposal for China is:
Historically, there have been many two character Chinese names - a few of them even still exist.
The new character could be added before or after the existing one. After promulgation of the name law, a period of one year would be allowed to make the choice.
The decision would be up to the individual adult at the time - not based on the head of the family (although most families who still remain in contact with eachother would probably choose the same name)
Choices would be limited to the characters already accepted in surnames.
To simplify matters, on the new ID cards, the old character would be underlined (for the first 50 years) - but the new character would be official & a required, real part of the name.
This would have to be handled very delicately.
Some people may choose politically sensitive characters - but, it's only a character. Let them choose it.
When celebrities announced their choice, many people would probably follow them.
And, for each existing common name there'd probably be one "obvious" choice on how to extend it, which many people would choose. Nevertheless, there would still be a lot of new, interesting names.
On the other hand, there might be a register built up of people's choices - and unique families might want to choose unique names. So being the first to officially register one of the unusual names would be like staking a claim on it.
Overlapping with someone due to this random chance wouldn't actually be so random - because actually, in the past, you shared the same last name with them anyway. (unless their name was A and they chose to rename themselves AB, and yours was B and your family chose to rename to AB - but in that case, your two families are made for each other anyway)
Along the edges of families, there might be trouble - one grandfather wants to choose one character, and all his descendants agree - but his brother wants to choose another, and his descendents agree, too. The choice might push the sides farther apart.
But the benefit of that is that it'd push the people with the same name closer together.
Chinese names nowadays usually have three characters - surname, plus two parts of the first name.
If the first section of a personal name within a family became standardized, then over time the family would actually acquire what is virtually a second character to their name. If this practice became standard, then this common character way to distinguish between distinct families of the same surname.