Why is 儿 in different positions in "儿子" and "女儿"? 儿 [Er] child  


  儿子 [Érzi] son
    女儿 [Nǚ'ér] daughter  


So, why is 儿 in different positions in each word but they are both valid?
This is an interesting question, because it allows us to look at how words are formed in modern Chinese.

The Characters

Both 兒 and 子 meant "child" or "son" in ancient Chinese. 兒 was more specific, while 子 had a variety of other uses, like "master" (as in 孔子 - master Kong/Confucius). When 子 meant child, it was somewhat inclusive of female children, although it can also refer to males only, as in the sentence 无子,有女二人 [he had] no sons, but two daughters.  

兒 can sometimes be used for "children" in general, as in the modern 兒童, but in ancient Chinese, it tended to be male-specific. The Qin dynasty proto-dictionary 倉頡篇 Cangjiepian gives the definition:
男曰儿,女曰婴 male [young children] are called 兒, female [ones] are called 嬰.
When 兒 refers to the child of someone, it means son.

女 means daughter, but has always also meant "woman" or "female (human)". 

Forming Two-Syllable Words

As the pronunciation of the Chinese language changed over the centuries, many one-syllable words became homophones. In order to clarify which word they meant, people formed two-syllable words that were less likely to be confused. Three very common ways to do this are:  


putting two words with very similar meaning together, as in 房屋 (house-house), 幫助 (help-help) or 美麗 (pretty-pretty)  
adding a more general word that clarifies the type of thing being described:  偉大 (grand-big) or 月亮 (moon-light)
only for nouns, adding 子: 桌子 (table), 梨子 (pear) and 褲子 (pants) all mean the exact same thing with or without the 子


It’s easy to see how 兒子 follows all three of these rules. I suspect it was formed by the third rule, because the 子 is pronounced in the fifth / neutral tone, which suggests it does not carry the meaning of the word.

So why 女兒 instead of 女子? Firstly, 女子 can mean daughter or "female child". But it can also refer to "women" in general. So 女兒 is more specific.
And why not 兒女? The word "daughter" would not be formed this way, because 女 is more specific than 兒, which can be gender-neutral. Therefore, 兒女 reads as "sons and daughters", which is indeed what it means.