The idea that I would randomly choose, or try to push the responsibility to someone else doesn't sit well with me. I had to work for about a decade before I started getting real responsibility for architectural decisions. I would have a hard time throwing that privilege away for brownie points or to "let someone else take the fall." No guts, no glory.
That said, some projects are Doomed to Fail. In which case, I see it as my job to become a thorn in the side of the person or team pushing for the project in order to get it canceled. On one such project, I would meet with my manager every single day and say, "You know there is no business reason for doing this?" She would say, "I know, but we still have to..." By the end of the first month, the meetings were very short. "You know what I'm going to say?" "I know." But after about 2 months, she had gotten the project canceled. 1-5 programmers had worked on that for 5 years. I felt very proud that I had succeeded where so many others had failed.
Sometimes, asking the questions that would encourage the business sponsor to give up or cancel the project teases out some expectation or aspect of how the software would be used that makes the project solvable, or maybe lets you come to a compromise that brings the possible software solutions back to the land of reality. Sometimes it gives me the necessary business understanding to justify a somewhat limiting or costly architectural decision, or a willingness to take on massive technical debt in order to keep the business running.
The sub-conscious mind is a powerful tool. If you have the problem loaded into your subconscious, then time can do wonders. Often a solution will just pop out when you least expect it. Review the problem whenever you feel your subconscious has moved on to digesting other things.
The more ways you can learn to think about a problem, the more likely you are to improve your potential solutions, so talking to people is a great idea.
Obstinately keep on looking for alternative solutions.
Sometimes this pays off, but sometimes this is a trap that makes you mentally spiral into an unhealthy place while getting nothing done. If if you feel you are just banging your head against a wall and you have the option to take a break, definitely take it (work on another project for a few days or a week), let your subconscious do the heavy lifting for a while, and come back to the original problem after some time has passed.
One thing that I have not seen other people suggest is to ask yourself two questions:
Which technical solution is most compatible or most flexible with the ways the business might use the software?
What kind of software are you building? Are there common themes that fit with one answer better than another? This may be a question about the business direction of the company.
The multiple-bad-solution situation moment is also the opportunity for the greatest creativity. It forces you and your business partners to ask the essential questions that define your software and your business. There is usually a "win" situation hidden in the moment of deepest darkness. If that's not the case, then maybe it's time to update your resume.