Budgets are being slashed throughout the world of former TV shows. You’ve already had to eliminate one of the cast of Friendsand now you have to eliminate one of the regulars on HIMYM. Try to make your answer as steak sauce as possible. Go!
Nice knowing ya Lily-pad.
Only one? 
Definitely Ted. I can barely remember he was on the show. There was some plot point about him, but I’m just drawing a blank. Nobody needs Marshall either.
Ditto. Some characters are funny, some insufferable but funny, but she’s just insufferable. Ciao!
All except Agent Hill and Dr. Horrible, and maybe the baker who moved to Germany if they could find a way to make her character more interesting than wallpaper paste. Everyone else, including Your Mother and The Kids, needs to be eliminated with extreme prejudice.
Stranger
Lily is IMO the least substantial character in the entire show, her backstory extends only to her relationship with Marshal and her eccentric father. Although I must say that I find it rather difficult to imagine Marshal without his spouse.
Robin. Lily had good chemistry with the other characters, even if she didn’t have much to do on her own. But Robin didn’t really have much personality, she was just there for Ted/Barney (and sometimes Lily) to moon over. And whatsherfaces acting is pretty meh to.
Kill off Robin, and make Lily a teenage Canadian pop-star so you can save the one funny Robin bit.
But then you lose the Twin Peaks references. Unacceptable.
Stranger
I voted Ted but now want to change it to Lilly.
Killing off Ted ends the show, so that’s a non-starter. And while everyone talks about Lily, I found that Marshal was the more boring of the two. Robin is funny, and Barney–who is going to remove him?
So, yeah, Marshal.
Definitely Robin. Yes, it was always going to be her (and the Robin Sparkles thing was the most hilarious thing ever). But her character was mostly meh.
Robin.
It’s a shame, 'cause I like the character (especially later as she got tougher and more Canadian) and the actress, but the only way to prevent the utter disaster of an ending is to kill her off, early S1. Ted meets her, steals the blue french horn, says “I love you” too early, things happen, Ted meets whatshername–the cupcake lady–much earlier, Robin slips in the shower, dies and is eaten by her 5 dogs that get sent to live with Robin’s aunt and her “friend” (quote-marks by Robin, not me).
The downside to this is that Robin/Barney made a brilliant couple and the two had some great chemistry. But to prevent the ending of the show? The sacrifice is worth it.
ETA: Yeah, and Simplicio is right: Make Lily the teenage Canadian pop-star.
It was hard to decided between Lily and Robin. Robin was just kind of there most of the time and didn’t really have any chemistry with the rest of the cast. Lily the character was an integral player but Alyson Hanningan really started to get on my nerves the last few seasons. So much so in fact that I stopped watching and didn’t even see the finale.
What four human abominations voted Barney off the show?!? I spit on your noodles!
Lily. Her character was the least integral to the show, when her role was featured the episode kinda sucked.
Hell no to Robin. That girl is hot.
Ted is the reason for the show (it’s in the title), so even though I found him the most frustrating, he has to stay.
Barney is the breakout star, so he stays.
Marshall and Lily individually aren’t that much, but they both play well with the group and the pathos of getting rid of half a couple doesn’t seem like it would fit the show.
That leaves Robin. The bonus is, getting rid of Robin means Ted and Barney quit mooning over her and can do something interesting, and Ted can have a decent ending instead of “But I always pined for the girl with the blue French Horn… wait, you’re mom’s dead, mind if I go bang her?”
First of all, STOP HATING ON LILY! I like her, she’s critical to the show, she’s the mastermind behind every plot on the show.
Secondly, Ted is useless. He doesn’t meet Teh Mother until the end, what a waste of time having him play a pointless part in all the other episodes.
Robin, is Canadian, have to keep her, everything Canadian is comedy gold.
I never liked Barney. He’s the kind of guy I would avoid in real life, while the rest I would dearly love if they were my friends.
I voted for whiney-ass Ted. He could have easily been killed and used only as a ghostly voice over.