Why to abandon an idea
In considering the question of whether or not you should write an idea you don’t like, my instinct would be no. If you don’t like it, it will show in the writing. If you are bored writing it, you can guarantee that your reader will be bored reading it. How excited you are by a project always comes across on the page.
Why to press on with an idea
All ideas start as little seeds. You have to give them time to germinate, you need to feed them with research, and give them time to grow.
Natalie Goldberg (Writing Down The Bones is an excellent book for new writers and I highly recommend the audio version) calls this process ‘composting’. You put in all the rubbish and wait for it to turn into beautiful soil.
How to develop an idea
I doubt many writers start out with fully-fleshed out stories. I certainly don’t.
I use Scapple from Literature and Latte to start a novel. It’s like a giant corkboard on which you can pin things. First I’ll pin my nut of an idea, perhaps a working title, maybe an idea for the theme.
If I can picture the characters in my head, I’ll download photos of them (actors and actresses I think could play them in the movie adaptations :) ) and pin those to the board. If that’s all I have, I’ll save the file and leave it.
As Natalie suggests, I give the idea time to compost, allow the seed to germinate. I’ll think about these characters just as I’m about to fall asleep. I’ll let them follow me around all day and see what they do, what they might say, whether they’re good or evil, and if they’re evil, what made them that way.
Each new idea, no matter how small or seemingly insignificant, I’ll pin to my corkboard (and I do this well in advance, while I'm still drafting a previous novel because I know it takes time and you can't rush the muse. She's like a cat; she comes when she feels like it, but most often when there's food around).
So, I feed the muse with research. For example, I barely have anything for my next novel, just a nut of an idea. But I do know that part of it will take place in a high-security psychiatric hospital. So, I feed the muse by researching hospitals, searching for case reports from psychiatrists, reading books like The Man Who Mistook His Wife for a Hat. This research may not provide anything useful but that doesn't matter, all I'm doing is feeding the muse, getting the cogs turning and giving the idea space to grow.
When my Scapple corkboard is fleshed out, I then turn to The Story Grid, which is an excellent resource for developing novels. It’s really about editing existing works but I use it to guide my fleshed-out ideas, create scenes that turn pages, and build a solid framework for a novel.
Don't give up
Out of all the ideas you have had so far, pick the one you're most excited by (even if that's a mediocre amount of excitement) and feed the muse. You may find that it's not so awful after all, and that with time, it develops into something rich and exciting.
If you stop feeding and watering every seed before it even germinates you will never grow anything.
BUT! I would recommend that you don’t start writing until you’re excited about the concept. Write short stories as Secespitus suggests.