My Thoughts
You don't want to look a gift horse in the mouth. What does that even mean? Think of it this way: if someone was polite enough to give you your very own horse as a gift, would you then inspect the horse to see if it is of good quality, especially when the person is standing right there? Of course not, that would be rude! Basically, be happy with what you got, and don't look for more.
There were those who would inspect the horse, particularly the mouth region. The reason for checking the mouth was to determine the horses age. A horses' age can be determined by how visible the teeth are. The more prominent the teeth, the older the horse is. Just don't expect the mouth if you received the horse as a gift!
Phrase Origins
According to Wikitionary, this phrase can be traced back to Eusebius Jerome, who was a famous translator of the Bible into the Latin language, and he lived in the 4th century CE.
Apparently, Jerome had used the common phrase in one of his writings, as he penned the words:
"Never inspect the teeth of a gift horse."
Interestingly, the expression seems to have the same meaning back then as it does today.
Examples of The Phrase Being Used
"My mom gave me a bunch of darker colored shirts as a present. Honestly, I prefrer to wear lighter colors, but I shouldn't look a gift horse in the mouth."
"Jacob looked a gift horse in the mouth when his wife, Amanda, bought him a powerful computer last Friday. His lack of appreciation for the offer of kindness was shown by his response: 'does it come with a printer, too?' "
Note: For most popular sayings and other common phrases you come across on this site, finding the precise phrase origin is difficult. What's provided is around the earliest known date the expression was being used in.
Usually the expression comes from a quote found in an old newspaper, book, poem, or play. However, if the idiomatic phrase is already being used in one of those, it's likely already a well known saying, thus it should be assumed the origins are older.
There are rare cases where determining an idiom's origin is possible. Anyways, for the most part, you'll get a rough idea on how long ago certain phrases were being used in, and can get a rough idea on how old they are.