Probably the best way to start coming up with short story ideas is to find out what sort of ideas turn into short stories. Check out http://asimovs.com/ and read the Hugo and Nebula nominees. The featured stories are good too. There's a link on that site to the Analog nominees as well.
If you find those stories a bit intimidating (as I do), you might want to read something of slightly lower caliber. http://dmoz.org/Arts/Literature/Magazines_and_E-zines/ is a good place to look.
Best of luck to you!
Then, Will Shetterly (an author you may have heard of) told me to get a copy of Rust Hills' book WRITING IN GENERAL AND THE SHORT STORY IN PARTICULAR.
Something clicked when I read that book, and I've been writing short stories ever since. I've sold six of them, and the first thing I wrote after I read that book was the first of the six.
The main thing to remember with short story ideas, as opposed to novel ideas, is that a short story is not only short, it works best when it deals with ones. One character, one setting, one day in time, one goal, and so on. (This is called "unity" and it makes writing short stories much easier--though they're still not easy.)
Read Hills' book and think in terms of unity, and see what you can come up with.
I think unity is the best word for what I was going to describe as what is going on when I've been able to write short stories. I have taken something like "what if my dog was immortal?", "what would a cyborg feel like once he became obsolete technology?" or "what if I could time travel and hear what x 'really said.'" I'm sure someone else could turn those into novels, but I was just looking at that one aspect of the issue and it came out to 5-20 pages. Some of those might later become the outline or part of a larger story, but for now, they are short and personally some of my favorite products.
What Hills talks about in his book, WRITING IN GENERAL AND THE SHORT STORY IN PARTICULAR,
sounds to me to be a lot like what Orson Scott Card talks about in his M. I. C. E. breakdown of story types as the character story.
Character stories work especially well in the short story format because they are about one thing--role changes.
If you approach your story idea as a role-changing problem for the main character, you will have a structure that will give you the middle and ending you may be searching for.
Taking Jackonus' idea of the immortal dog, for example, could give you a story about how the dog's role of being someone's pet would change when the owner died. You'd probably need to bring another idea into that (maybe add the cyborg with obsolete technology, and have them find each other?) to get it to work, but you could write a short story about how the dog's role and/or the cyborg's role have changed and what they do to either try to find a new role, try to get the old role back, or adjust to the lack of a role entirely.
(Actually, looking at Card's M. I. C. E. breakdowns can help you figure out the structure of whatever kind of story you are writing. Be aware, however, that milieu and event stories tend to work better at novel lengths, and idea and character stories may be more suitable to shorter lengths.)