Found this definition on Luciferous Logolepsy via Onelook.com:

"accessit
n. - honourable mention of one who comes nearest to a prize."

More obviousizing: So, let's say there were three prizes: first, second, and third prizes. The accessit would be the honorable mention after the third prize, right?

A pretty useless word since everybody so freely uses honorable mention. Who needs another word to rub it in that you didn't get a prize?