"子" (or "こ") often comes at the [end of a woman's
name](http://www.sljfaq.org/afaq/names-for-people.html). But for baby animals,
"子" is at the [start of the term](http://www.language-
exchanges.org/es/content/confusing-about-animals-name). Why is it different
for the two?

(I've heard of the name "Koichi" belonging to a Japanese man, but I don't know
if that literally means "child one", uses "子", or is cognate with "子")
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grammar names
Why does "子" often come at the end of a woman's name, but comes at the start of the term for a baby animal?
This is difficult to answer, but:  
TL;DR **Because the meaning of 子 in the end of name and that of 子 at the start
of term meaning baby animals are different!**

* * *

"子" (or "こ") as the end of a woman's name and "子" as the start of the term of
baby animals (I think you mean something like "[子猫]{こねこ}".) What "子" here
makes are very different.

  * 子 as the end of a woman's name **"makes a name."**
    * This is a **suffix**. 
    * According to [this site](https://ja.wikipedia.org/wiki/%E5%AD%90_\(%E4%BA%BA%E5%90%8D\)), 子 as suffix of name is seen way before Heian era, and has been used in man's name back then.
    * Looks like that in Heian era, the emperor gave the princess the name so: they first replaced the kanji in their name with the one of same reading (because they looking bad,) picked one of them (name back then contained multiple letters,) then added 子. 
    * Then the name using suffix 子 became widespread.
  * 子 as the starting of the term of baby animals makes a "term." 
    * In this case, it works as a _prefix._
    * This 子 has the obvious meaning of "baby of."

And no, we never say baby cats as "猫子." It's weird. Using 「子」 as prefix,
babies of animal (grammatically `general noun`) are often called "子-(name of
animal.)" For instance, 子猫{こねこ}、子犬{こいぬ}.  
But also note that there are some cases that "子-prefix" sounds weird.

* * *

> I've heard of the name "Koichi" belonging to a Japanese man, but I don't
> know if that literally means "child one", uses "子", or is cognate with "子"

It doesn't. Koichi is read "KO-O/I-CHI" ( **こう** /いち) and written as 光一, 幸一,
浩一 etc.
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answer
If you'd really think about it, in English, it wouldn't be as confusing as it
seemed.

For example, when we say "baby kangaroo", we're referring to a "kangaroo", a
small, young, " _infant_ kangaroo". We're using a noun _baby_ as an adjective
here. Generally, nouns can be used adjectively in English if put before
another noun: " _ice_ age" actually means "glacial", " _winter_ bash" is
"hibernal" (thanks, @oals). So does Japanese. `子 + animal` means "baby animal"
**because** it's put before the animal name.

In personal name ～子, the 子 is but an ordinary noun _child_ (or, _girl_ ). Thus
`words + 子` just means "--'s child" or "a child of --" or "a -- child".
Originally this naming convention was meant to be "a [adjective]
(noble-)descendant", and why it becomes female name maker is a long history,
please see @CollapsedPLUG's answer and link.

**PS**  
～子 compound isn't limited to woman's names. It also has common usages:

  * just as "[something] child":  
みなし子【ご】 "orphan", 継子【ままこ】 "stepchild", 赤子【あかご】 "( _old-fashioned_ ) baby",
双子【ふたご】 "twins" etc

  * or more sporadically as "the [something]-ed" or "[something] worker":  
売【う】り子【こ】 "(extra) vendor", 切子【きりこ】 "rounding-off; cut glass", 踊【おど】り子【こ】 "(
_old-fashioned_ ) showgirl", 店子【たなこ】 "tenant" etc.
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answer
The origin of languages and customs are often forgotten and we often just do
it without regard. Most Japanese people don't know the history about 子 as
stated above. As I understand it, 子 in a name, has a cute sound to it and for
a few decades it was popular to use. --Think of it as calling someone Billy,
Bobby, Nicky etc.
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