He was one of the three finalists, along with Eugene H. Krabs and Huford BargainBottom. In his video submission, titled, "Tying up Loose Ends", he is seen in the restroom collecting stray pieces of toilet paper, and then tying them back onto the roll. His name is a pun on the word, "tightwad", which means someone who is really cheap.
Oddly, Dwight T. Wad was technically the only one whose video displayed cheapness (as cheapness is an unwillingness to part with money). Eugene H. Krabs and Huford BargainBottom's actions would be considered cheating money out of customers, while Dwight T. Wad was actually being cheap when he tied together the old toilet paper pieces, rather than throwing them away.
Wad also appears in "Hello Bikini Bottom!" as a supermarket manager. He makes a bill for Mr. Krabs for having his band play at his store. He appeared again at the end of the episode to collect his money from Mr. Krabs.
It is possible that the surname "Wad" is a reference to the common phrase "A wad of cash," meaning a bundle of money.
Also, the "T. Wad" portion of his name may be a reference to the phrase "tight wad" which means a cheapsake.