I have a flight reaching Houston International airport at 14:00 hrs. I have a connecting flight within 1hr and 30 mins. If I miss this flight due to immigration and customs delay, I would like to know the cheapest means to travel to Lafayette from Houston International airport?Are there any alternate routes?
Given you've said that this is a connecting flight I'm presuming that both flights are on the same ticket.
If you miss the connecting flight, even if it's due to delays in customs and immigration, then the airline will change you to a later flight at no cost. Simply go to the customer service desk at the connecting airport and they will look after you.
If the next flight is the following day, and the problem wasn't their fault (eg, the inbound aircraft wasn't delayed) then any additional costs such as hotels will be your responsibility. If the inbound flight was delayed then they may or may not offer to cover a hotel/meals depending on the specific circumstances.
Even if the flights were booked on separate tickets it's worth checking with the airline to see if they will re-accommodate you on a later flight. For example, United Airlines has a "flat tyre" policy that as long as you arrive at the check-in counter within 2 hours of your flight departing they will attempt to put you on a later flight at no charge.
Rome2Rio shows three alternatives to flying: bus, train, and car. The bus is a Greyhound service that takes 4h43. It departs from Houston city though so you'll have to find a way to get to the bus terminal. The train is an Amtrak service that takes 5h06. It too departs from Houston city and will require you to find a way to get to the train station. Finally you could rent a car and drive, of course.