What traditions do people follow when naming a second child in Chinese?
First I'll state that you can use whatever Chinese names you want; unlike western names which often have standard names from the bible for example, theoretically you can use any combination of the thousands of Chinese characters to form a name.

This means that there are so many choices that it's hard to decide on the name, so most parents use some convention or system to help them pick names. Nowadays the traditional naming systems have long fallen out of favour, so there's nothing stopping you from using any convention, system, or no system for the names.

https://en.wikipedia.org/wiki/Chinese_given_name

Same character for siblings

A common convention, as you noted, is using the same character in the same position for two-character names. The repeated character is usually in the first position.

Splitting words

Another common convention is to name siblings by splitting a word or idiom. An example would be naming the first child 健 and the second child 康. You could combine this convention with the same character convention.

Generation names

A tradition that has largely fallen out of practice since Republic times is the use of generation names - every sibling in the same generation would contain a certain character, and the characters would often form a generational poem. For example, Mao Zedong's family had the following poem, and he was the 14th generation: "立显荣朝士，文方运际祥。祖恩贻泽远，世代永承昌". Note that Mao Zedong himself broke with this system in naming his children.

Birth order in courtesy names

Another tradition that has also become archaic together with courtesy names is the use of special characters to denote birth order within the same generation. These are bó (伯) for the first, zhòng (仲) for the second, shū (叔) for the third, and jì (季) typically for the youngest. For example, Confucius's courtesy name was Zhòngní (仲尼), meaning he was the second son.

I believe that the generation names and birth order names are the only proper naming systems, which influenced the conventions used today. For example, the use of generation names would give the appearance of using the same character for all siblings, but the repeated character is not arbitrary. These naming conventions are also used with daughters, when traditionally these systems were only used for sons. In other words, these conventions are not followed strictly at all, so you can use any convention or system you like. Unless you're superstitious that is, in which case you could try a literomancer.