A ratio is effectively a fraction of the form m/n where m and n are integers. The fraction is therefore a rational number. So it follows that an irrational number is a number that cannot be expressed as such a ratio.
Rational in other contexts means reasonable, logical, sensible, commonsense,... So irrational implies the opposite: unreasonable, illogical, nonsensical, ...
It may have been assumed at one time that all numbers could be expressed as ratios or whole numbers. That's a reasonable assumption. But then numbers that didn't conform to this expression were discovered, numbers like √2, etc. This family of numbers had to be given a name, so what better than irrational?