I participated in a weekly challenge prompt app and many asked me to continue writing and to make it a book but I can't find a good idea to develop that short story.
Warning: This answer is very long — much longer than I was expecting it to be at any rate, so may take some time. Hopefully the content justifies the length.
Short story ideas are inherently different from novel ideas, so it may not be possible to adapt one into a full-length book unless you have a sure-fire direction to take it. Such an exercise would also depend on the genre of the story — certain genres like literary fiction or horror would be considerably harder to adapt than, say, fantasy. In general it's wise to begin by editing your story (short stories don't have a lot of fluff, so pay attention to word choice and sentence structure, which are far more important in shorter writings).
Secondly, it's important to be aware of what theme you are exploring in your short story, since shorter writings do not attempt to tell long tales. They focus on a few characters and, at most, a couple of events.
A lot of people believe that short stories can be taken as prompts for novels. This is untrue, since if the part of the tale covered in the short story appears at all into the novel, then it will probably not be the beginning or the climax, but rather a scene in the middle.
That said, if you're hell bent on making it a novel, consider:
Table the story for a few weeks and read a variety of different books, especially those very different from your genre, and combine the influences.
Take a look at these two loglines. The second one is from a variety of influences, and is more adaptable into a full-length novel (it's actually derived from the plot of Thomas Pynchon's The Crying of Lot 49).
Following a mix-up at the post office, a PI attempts to retrieve two letters which form essential evidence to a high-profile murder case.
Let us say that this is our short story. It begins with a visit from a housewife and quickly ends with the PI recounting how he solved the case with onlookers remarking at his cleverness. Very Sherlock Holmes, but the idea wasn't enough to warrant a full-fledged novel. So we end up reading Umberto Eco, do our research, and come up with this:
A paranoid housewife accidentally stumbles across two letters which may be essential to a high-profile murder case and (with the help of her LSD-fuelled psychologist) ends up unearthing a vast conspiracy in the postal system stemming from a centuries-old rivalry.
It's interesting, different, and a subject which cannot be covered in a short story. Whereas the short story was a fairly straightforward murder case, this one devolves into a much more fleshed-out mystery with considerably more charismatic characters.
- Exploring Character Backstory.
Since a short story covers only a few events, there is often some sort of backstory leading up the the canonical events. For example, several of Joyce's stories in Dubliners involve backstory, most notably Eveline (in which the protagonist recounts her neglected childhood while mulling over an ethical dilemma).
In several cases, it is possible to draw inspiration from the characters in your short story and flesh them out.
As before, we can look at a simple logline:
Sir Lancelot is pitted against a dragon that has a nasty habit of hoarding neighboring princesses.
Here Lancelot is a larger-than-life character, so it's easy for him to have a colorful backstory which can be exploited. Lancelot is on a mission for his king, which implies that the king sends him on numerous missions. That could also mean that there are more knights employed by the king apart from Lancelot, and after some creative thinking, you decide on the name Knights of the Round Table.
An ageing Sir Lancelot recounts the various adventures of the knights of the round table, an elite group put together by his prodigious liege King Arthur for a God-given quest to find the legendary Holy Grail.
- Expanding plot threads.
Unless your short story focuses heavily on a single character, many of the intertwining plots that form the base of the story can be fully fleshed out. This is a slightly riskier method than the other two because you're not incorporating more events into the storyline so much as building those already in place.
The story of a dysfunctional family with a tragic ending is told through innocent excerpts from a five-year-old's diary.
This is the main plot of Pikoo's Diary, a short story written by Satyajit Ray. By building on some of the central conflicts of the story (his mother's adultery, the subsequent fallout between the different members of the family, father turning to drugs, etc), there can be several additional plot points — although it's possible that might come at the cost of the entire schtick of the story.
Going though a messy divorce and determined to take custody of his five-year-old daughter, a sports journalist looks back at his own parents' failed marriage through the pages of his old diary.
Here we add a parallel and transfer the plot points of the diary to the frame story (which is recounted by the adult narrator in first/third person), and instead use the diary as character motivation.
These are some of the ways you can gain ideas for expanding your story.
Keep in mind, however, that if you're having trouble naturally coming up with ways to transform the story then perhaps the tale is better suited to that medium. There's nothing wrong in a short story, after all — many of them have emotional cores much deeper than novels, and the medium also allows a much greater deal of experimentation.