I don't think Luigi's a coward. He's a NITWIT, but certainly not a coward!
A real coward would abandon his loving brother all together to a group of ghosts and could never free him! (Luigi's Mansion)
A real coward would abandon his loving brother all together when he feels too stiff about the rumours he hears of a monster that eats people, even when that monster holds the vital item to cure the coward's brother from his illness! (Mario & Luigi: Superstar Saga)
Luigi is a big, fat idiot, who doesn't how brave he really is. All the scary things he thinks about is only scary in his mind!
Luigi can be quite a clutz - very clumsy indeed! Luigi may be tend to be pessimistic about the situations, his whereabouts, and even the adventure as a whole. Luigi may not recognise things clearly until he looks at it thrice. Luigi may claim very unlogical assumptions. Luigi may try to loaf around during the worst of times. Luigi may tend to cry a lot. Luigi may even say the worst of jokes. Then Luigi can certainly not know how brave he really is most of the time!
Luigi can obviously show he's a very scared person, but that does not mean that it makes him in any way a coward!
Despite his moronity, it is his very bravery which makes him the best sidekick a big brother could have!
Well, he's not perfect, but still a good sidekick...
Let's say Luigi is more of the Fun Side of the Brothers!