Nsryjdtyk

Examples would be like someone saying:

“Well, today was awful! I can’t believe that just happened to me!”

or

“Wow, today is great! I’m so lucky!”

...to get others to ask what happened.

What is a common term for this?

Answering/naming the activity (rather than labeling the person) is usually more artful use of English and more effective in making your point. Consider the terms "baiting" or "fishing".

bait:

verb

8. (transitive) to entice; tempt

fish:

verb

17. 
    

    (intransitive; foll by for) to seek something indirectly

    
    
    

    to fish for compliments

    
    

^{definitions from Collins dictionary, English section}

Examples:

"Stop baiting people into asking you questions."

"You're fishing for questions. I like you, but you won't catch any today."

You might call it "question baiting" or "question fishing".

To answer the question fully, and I do not mean any sarcasm at all, if you must have a label for the actual person doing this, the urban dictionary would probably be best suited, partially because vernacular might be more effective there and mainly because we avoid single-word answer questions on ELU.

When done on social media, it goes by the name vagueposting. You could generalize to vaguetalking.

vagueposting

1. Give off a vague feeling of sadness or other emotional issues to prompt others into asking what's wrong and generally shower said poster with attention

Urban Dictionary

Statements of that sort could be called teasers, defined variously as:

a ... device intended to arouse interest or
curiosity especially in something to follow (M/W)

or

a person or thing that pokes fun or raises interest, without
intending to satisfy (yourdict)

I don't think there are any perfect matches. Still, here are a few

They are 'piquing someone’s curiosity' or 'piquing their interest'

to make someone want to know more about something or someone.

Example:

Eventually her curiosity was piqued and she went to talk to the boy.

Macmillan

I think 'piquing their interest' might be the best recognised phrase, even better when prefaced by 'deliberately', as in 'he was deliberately piquing their interest'.

They might be 'drawing someone in'.

If you draw someone in or draw them into something you are involved with, you cause them to become involved with it.

Collins

It's not a phrase I've heard much or at all, but to 'manufacture interest' is probably as good a fit as I can think of - as good as piquing interest, since it is clearer but less common.

On a related note there's also to 'arouse interest' or 'raise interest' but I don't think those are quite right, and I could not find any good source links for them anyway.

A specific version of this is called the pronoun game. This is used when the "little information" given is an unspecific identifier such as "he", "she", "it" or generic terms like "there", "that", etc., forcing the other person to ask "who" or "what" or "where, exactly".

(note: The term has also been usurped in a genderism context, I'm not referring to that meaning)