What was President Wilson's biggest mistake in intervening in the Mexican War in 1914?
It is not possible to say objectively that President Wilson made any mistakes in his policy towards Mexico in 1914. You can argue that he made mistakes, but you cannot objectively prove that he did. Any statement about which of his actions was his biggest mistake is simply a matter of opinion.
One possible view is that Wilson’s biggest mistake was probably his actions in the Ypiranga Affair. In 1914, Victoriano Huerta was ruler of Mexico, having seized power in a coup. The US disapproved of the way Huerta had come to power and it disapproved of his dictatorial rule. Therefore, Wilson supported Huerta’s opponents in what was essentially a civil war. When Wilson learned that the German ship Ypiranga was bringing arms to Huerta’s government, he ordered US forces to land in the Mexican city of Veracruz and prevent the weapons from landing. The attempted occupation of Veracruz led to the deaths of at least 150 Mexicans and soured relations between the two countries. One could argue that this was a mistake because the shipload of arms was not important enough to warrant what was essentially a US invasion of a foreign country.
A second possible view is that Wilson’s mistake was to demand too much from Mexico in the Tampico Affair. This happened when Mexican authorities arrested American sailors in Tampico. The sailors were quickly released, but Wilson demanded an apology and that the Mexican government fire a 21-gun salute to the US flag. The Mexican government apologized, but refused to fire the salute. You could argue that it was silly for Wilson to demand a salute because the salute was not important in any way. You can argue that Wilson essentially pushed Huerta into a corner by being too demanding.
Either of these can be seen as Wilson’s biggest mistake, but there is no way to objectively say which one was the biggest mistake.