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犬子 (literally, dog + son) is an archaic, self-depreciatory term to refer to your son (occasionally it's used to insult other people's sons).
This answer raises a question though; who is the dog referred to in the word, the father or son? Both explanations work for the purposes of self-deprecation, and most dictionaries do not answer this question because the word is fixed.
The dog refers to the son. The term 犬子 originally meant "puppy":
【漢·列仙傳·邗子】邗子者，自言蜀人也，好放犬子。時有犬走入山穴，邗子隨入。
So calling one's son 犬子, would have been in essence referring to a child as "my little pup". That was not originally a self-deprecation. Instead, it was a childhood nickname for a famous poet, Ssu-ma Hsiang-ju:
【漢·史記·司馬相如列傳】司馬相如者，蜀郡成都人也，字長卿。少時好讀書，學擊劍，故其親名之曰犬子。
This "puppy" went on to become a literary icon of Han poetry. As his fame grew, people began imitating him, and started calling their own children "puppy" after Ssu-ma's childhood name. As that practice became more widespread, eventually 犬子 became a standard deprecation for sons.
In summary, the answer is that the dog in 犬子 refers to the son. Of course, you could also say that calling your children a puppy always implied that you, as their parent, was a dog.
I think the dog refers to son. Chinese parents called their sons "dog" mostly because they hope their kids can easily grow up as puppys. Depreciatory is also an important reason.
the dog is meant to describe the son. 犬子 means a boy like a dog. We think dog is not a powerful animal as opposed to tiger or lion. You can get the feeling in this phrase "虎父无犬子", meaning a father like a tiger can't have a son like a dog, which is usually used to compliment other person's son and their father. On the other hand, calling your own son 犬子 does imply that yourself is a dog. Similar expressions exist like 兔崽子, 小畜生.
谦虚的说法，介绍自己的孩子通常用 犬子.
《史记·司马相如列传》：“少时好读书，学击剑，故其亲名之曰犬子。”
Did some research, although not very authoritative it still seems quite convincing:
个人觉得应该只是谦称
毕竟中国人一直讲究谦虚，不可能见人就称自己儿子是虎子
大概人说虎父无犬子，所以才会有犬子一称来谦虚的说自己的孩子
网上只说明犬子其中一个意思是指司马相如的小名
并没有明确说犬子来自司马相如的小名
The basic idea being that the phrase came from: 虎父无犬子: lit. lions don't father dogs - or lion father =/= dog son - so in order to be humble fathers would refer to themselves [seems both father and son] as dogs.
Seems similar to son of a bitch where obviously both the mother and son are being referred to.
小犬、豚儿、豚犬 are all referred to as alternatives to 犬子 - I find the 豚字 particularly interesting. (pigs and dogs = sons).
Definitely the son. 虎父无犬子: Eagles do not breed doves. Opp: 上梁不正下梁歪: Fish begins to stink at the head. (上梁 refers to the father or leader)
