I think this problem actually requires two solutions:
#1. Users who admit they don't know where to put a question:
I think a 'global' location within the Stack Exchange organization such as:
http://stackoverflow.com/questions/ask
would be a good place to [a site recommendation system](https://meta.stackexchange.com/questions/98192/stack-exchange-various-site-organization-and-map/98251#98251). It would be an undertaking -- as George Edison notes in the comments -- but I'd be willing to help get it off the ground. Statistically, it would be an interesting classifier problem.
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#2. Users *who think* they know where a question belongs:
To address this -- the focus of the OP's question -- I suggest the following considerations:
* Parsing tags may be *the most useful* meta-data to decide if a question is on topic; however, suggesting that a question be migrated after adding a tag could become quite distracting. If new users are uncertain of how to tag a question, they may start typing simply to see the [new awesome tag suggestion popup](https://meta.stackexchange.com/questions/100669). Suggesting a site after each tag would become very distracting and negatively affect the user's experience. Hence, if the question belongs on a different site, **the migration recommendation should appear _after_ clicking 'Post Your Question'**.
* Providing a means to seamlessly migrate a question will lessen the burden placed on the moderators to migrate a question.
* A classifier system won't be perfect. So **it shouldn't force the user to migrate a question**.
* **Education is important**, so if this were implemented, I think the users should *see* the bulleted list of on-topic [from the faq](http://stackoverflow.com/faq#questions) for both the current site and the suggested sites so that they can *learn* where to place the questions in the future.
* **Migrated tags won't work**, so if the user decides to migrate a question, they should be prompted to retag a question as some sites use one tag naming convention and some sites user another.
So how about a page, much like the CAPTCHA page, which appears after a users clicks 'Post Your Question' and says (Please imagine radio buttons instead of [O]):
> *After parsing the content of your question, we believe your question might be more appropriate for a different site within the Stack Exchange Network. Please consider migrating it to one of the following:*
> [O] **Super User** ![right arrow][2]
> [O] **Server Fault** ![right arrow][2]
> *However, in our recommendation, we may not be 100% accurate, please feel free to post your question to Stack Overflow (the originally selected site) with the understanding that the community may migrate your question if they feel it would be a better fit and **receive a faster/more complete response** elsewhere.*
> [O] **Stack Overflow** ![right arrow][2]
> **Please select a site from the list above to post your question**.
Then if the user clicks the arrows, they would see expanded descriptions:
> *After parsing the content of your question, we believe your question might be more appropriate for a different site within the Stack Exchange Network. Please consider migrating it to one of the following:*
> [O] **Super User** ![down arrow][1]
> is for computer enthusiasts and power users. If you have a question about …
> * computer hardware
> * computer software
> and it is not about …
> * videogames or consoles
> * websites or web services like Facebook, Twitter, and WordPress
electronic devices, media players, cell phones or smart phones, except insofar as they interface with your computer
> * a shopping or buying recommendation
> [**Super User FAQ**](https://superuser.com/faq#questions)
> [O] **Server Fault** ![down arrow][1]
> is for system administrators and desktop support professionals, people who manage or maintain computers in a professional capacity. If your question is about …
> * Servers
> * Networks
> * Desktop PCs that you maintain in the workplace
> and it is not about …
> * Networking outside the professional workplace
> * Running servers at home for personal use
> * General personal computer troubleshooting
> [**Server Fault FAQ**](https://serverfault.com/faq#questions)
> *However, in our recommendation, we may not be 100% accurate, please feel free to post your question to Stack Overflow (the originally selected site) with the understanding that the community may migrate your question if they feel it would be a better fit and **receive a faster/more complete response** elsewhere.*
> [O] **Stack Overflow** ![down arrow][1]
> is for professional and enthusiast programmers, people who write code because they love it. We feel the best Stack Overflow questions have a bit of source code in them, but if your question generally covers …
> * a specific programming problem
> * a software algorithm
> * software tools commonly used by programmers
> * matters that are unique to the programming profession
> [**Stack Overflow FAQ**](http://stackoverflow.com/faq#questions)
> **Please select a site from the list above to post your question**.
[1]: https://i.sstatic.net/iNO7e.png
[2]: https://i.sstatic.net/rGk7d.png