LM: DM is actually a sequel to the first Luigi's Mansion for the game cube, something I admit I never played because I never owned the system. Although I watched a play through to get a better idea of what happened prior to this one. The first game was actually the start of Luigi's character development which was explored in various games ever since. While Mario is the fearless adventurer his brother...isn't. Luigi is more fearful and has to face his worst fear (ghosts) in order to save the day. It made him a more rounded character than Mario and well, far more relatable.
A few SPOILERS below for some older games, Mario & Luigi: Dream Team and Luigi's Mansion Dark Moon.
With the help of Professor E. Gads our green plumber was able to take on an entire mansion full of ghosts with the trusty Poltergust 3000 and earned the wrath King Boo. With his brother saved things mostly went on as normal. Only this time Luigi started to get more storylines hinting at or directly mentioning his insecurities of living in Mario's shadow and his fears. This was usually played for laughs to various degrees. The extreme cases making him try to flee at the first sign of trouble (*1) the more tame cases being when he expressed a moment of uncertainty. For example one of the played for laughs bits had him dressing in drag to pretend to be the princess to fool the bad guys. In another he'd keep a journal to write about his jealous feelings towards to his brother getting all the attention but still admiring Mario's courage. It usually changed how extreme it was based on the system or series he was being used in. But this wasn't all in his head because other characters tended to ignore him too often praising Mario then asking who he was. It got to the point even Bowser didn't call him by name often referring to him as the Green 'Stache.
From what I recall it was never outright explained what his relationship to the games doomsday device, the Chaos Heart, was. There have been hints in other games that Luigi might have some darkness to him or at least is hiding a dark secret. Whether that was the reason for his connection to the Chaos Heart or jumping on it at the start of the game...it's not clear.
But what's really a treat is when Luigi gets to be in the spotlight. In Luigi's Mansion Dark Moon the green plumber is called by his friend E. Gadd. Since Luigi is the best and only ghost catcher he knows he's need to capture the rogue spirits. What I like about game is the somewhat subtle plot points you might miss. He starts out helping this time around because he's needed not because of personal reasons. There are also some Toads throughout the game that need to be escorted back to E. Gadd. All of them have fears like Luigi usually freaking out around ghosts, water and in one case clocks. But it's up to him to get them through all of it and while doing so his animation has him acting very encouraging. Likewise the Toads give him some much needed affection and thank him for his efforts. Something that past games haven't done as some characters who aren't Princess Peach tend to look down on him.
Not to say Luigi doesn't still get some insults slung his way. The ghosts do taunt the player/Luigi through out the game and even E. Gadd makes some sarcastic remarks. But nothing's as mean spirited as previous titles although the title character still falls victim to comedic pratfalls and the like. Another thing that I was pleasantly surprised with was just how important Luigi was. His importance has been underlined in a few past games but always in the shadow of other characters and events. The focus is always on him here, more so than the titles he previous starred in Luigi's Mansion and Mario's Missing. Right from the start the player learns that King Boo is the one to blame for everything going wrong. The reason is that he's pissed at Luigi for trapping him in the first Luigi's Mansion and this is his revenge. Although Bowser has had his moments when he acts scary King Boo proves why he's such a great arch-enemy for Luigi.
He makes our scared plumber deal with his greatest fears, many of which are elaborate death traps and dangerous enemies. This is one of many things that makes the lead character more endearing. No matter how bleak things look Luigi still pushes through to save the day. It makes him more heroic for facing his fears and the character development is amazing. Then there's Mario and Luigi Dream Team where Luigi plays a vital role in the plot. Not only is his most personal thoughts on display he get one of my all time favorite Mario/Luigi game battles by taking on Bowser. The best part is that he impresses Bowser so much that the King of Koopas stops himself mid sentence to refer to Luigi by name out of respect. Luigi should get more focus because he's the ultimate underdog and that makes him the hero I'll always root for.
*1 From what I can tell this was only used in the Mario & Luigi series on DS but maybe it's in some titles I haven't played.