Someone may know more than me but :

Mexico was a big regional power and by installing a ruler French influences would be a plus. No matter who was installed by France, he would have " obligations " towards France.

Then, France would put in place a Catholic ruler in charge of Mexico a mostly Catholic nation. This would be a " wall " against the growing power of the United States.

Mexico was a country in terrible financial difficulties but was big and naturally rich. France would harvest the fruit of installing " its own " ruler when Mexico would recover and start to develop ( industrialize ). Similar in many points to the now US, Mexico could become very attractive for immigrants.

Plus, the US could not do much concerning Mexico at the time considering the state of the country.

Napoléon III searched several candidates among European princes ( Catholics ) but most of them refused or proved not " worthy " in France's eyes. Maximilian of Hapsburg was a good candidate but not attracted by the idea of ruling Mexico. His wife Charlotte though influenced him and he accepted, unfortunately for him when we know his fate ( a shame by the way, he could have been a great monarch if he had more time and better conditions ).