The most important thing to remember in analyzing literature of any kind—certainly poetry—is that authors choose their words for a reason. Especially when you're close reading, assume that the author has thought through every single word choice and picked the one that conveys his or her meaning as completely and precisely as possible.
The title might be the most important word choice of all, because it's the first thing the reader sees, so it shapes the way the reader understands the poem. In your analysis, then, the title is a big clue: it tells you what the author wants you to be thinking about as you start reading.
When the title doesn't seem to match the content of the poem, you have to ask yourself why the author wanted you to be thinking about something that isn't obviously relevant. Was the point to make you confused? Does the title add some bit of information that makes you see the poem in a different light? Do the words in the title have multiple meanings?
The most clever titles often seem to have one meaning, but by the end of the poem, you realize they have multiple meanings. Take "Darkness" by Lord Byron (you can read it here). It's a poem about what happens on Earth after the sun goes out. The title's meaning is obvious: when there is no sun, everything is dark. This is a poem about stuff happening in the dark. Duh.
But Byron imagines that, on a dark Earth, humans would begin fighting each other to obtain the few remaining light sources (wood to burn, for example). By the end of the poem (spoiler alert!), every human on the planet is dead. They all killed each other. So, when we circle back to the title, "Darkness," we realize that Byron is also talking about the darkness inside us. It's a commentary on what he sees as one of humanity's deepest, darkest flaws: we are selfish creatures that will do anything to stay alive, even if it means killing each other.
So when I'm analyzing a title, here are the questions I ask. Before I read, I ask: what does the title tell me right away? What does it send me into the poem knowing?
As I read, I ask: what's going on in this poem that relates to the title?
After I read, I ask: Why did the author want me thinking about the specific words in the title? Do the events of the poem make me understand the title in a different way?
And when I'm really stuck, here's what I do: I make up other titles for the poem, and figure out why the author didn't choose to use them. Why didn't Byron name his poem "The Day the Sun Never Rose," or "The Fight for Fire"? I think that focusing on the sun ignores what the humans are doing, and focusing on the fighting obscures his real purpose, which is to ask why everyone is fighting. He wants us seeing that the outer darkness revealed our inner darkness.
Here are a few examples of how I'd use those questions to analyze poem titles. I've tried to include a couple unusual ones.
"The Raven" by Edgar Allan Poe (read it here). At first, it seems pretty straightforward. The narrator gets visited by a raven. But why does Poe want me focusing on the raven, not the narrator? Why isn't the poem called "Once Upon a Midnight Dreary" (the setting and first line), or "The Reader" (that's what the narrator is doing at the beginning), or "Nevermore" (what the raven says), or "Lenore" (the narrator's dead lover)?
At the end of the poem, I realize that the big mystery is the bird's nature. The specifics about what it does (like saying "nevermore") aren't nearly as central to the poem as the question of what it is. Is it real? A devil? The spirit of Lenore? A hallucination? The title made me focus, in other words, on the identity of the raven itself, the big mystery, and that led me to wondering about why it came to the narrator and what's going on in his head. It guided me to the questions Poe most wanted me to ask.
"Composed upon Westminster Bridge, September 3, 1802" by William Wordsworth (read it here). Sounds pretty dull and extremely straightforward. I go into the poem knowing exactly where the poet is and when he's writing. It's not even telling us anything about the poem's contents, so why bother analyzing it? Here's the thing—why doesn't he want to tell me anything about the contents of the poem? Why does it matter that I know where he's sitting or standing as he writes?
The poem is about how beautiful Wordsworth thinks London is. So as I finish and circle back, I'm wondering why he didn't make the title something like "London," or "The Fair City." I realize that the title isn't just record-keeping after all—by putting readers on a bridge in London, he's making us feel like we're in the middle of the city too, surrounded by the same beautiful scene. He wants us to be seeing out of his eyes. If he called the poem "London," we'd start reading feeling like we were looking at the city from outside. It also gives me the sense that the moment was so profound he's saving the date and time the way we might write on a picture in a scrapbook—he'll always remember that September 3, 1802 was the day he saw the most beautiful thing ever.
"Porphyria's Lover" by Robert Browning (read it here). Right off the bat, I know this is a poem about someone named Porphyria and someone who loves her. Porphyria is a weird name, so I look it up and learn it comes from the Greek word for purple. So why should I be thinking about a purple girl and the (presumably) guy who loves her? In the poem, the lover, who narrates in first person, goes to see Porphyria, and thinks the moment is so perfect that he strangles her with her own hair, then cuddles with her corpse. This is an example of an unusual title, I think, because of the contrast between the title's description of the narrator and what the narrator actually does. Why call him a lover when he murders his girlfriend?? So now I'm focused on trying to figure out why Browning used such an apparently inappropriate word.
Why not call the poem "Yellow Hair," or "Murdered for Love," or "The Perfect Moment," or simply "Porphyria"? Clearly, Browning wants us going into the poem knowing that the narrator loves Porphyria. Is it a straight-up lie, just for shock value? Or is it possible the guy is truly in love with Porphyria? Or is he insane? And if he is in love with her, could he possibly have a decent reason to strangle her? Is Browning trying to tell us something about love itself by juxtaposing it with murder?