I am posting this mainly to summarize some points mentioned already in the comments - in an answer they can be a bit expanded and presented in somewhat better form. (The comment section grew quite long, I am one of the people to blame for that.)
I guess that at least some of the information mentioned here might be useful for the decision whether and how MO Twitter account could be set up.
Twitter account of other SE sites
As you can notice, many sites in Stack Exchange network have twitter accounts. Tweets are posted by twitter bots which were set up by SE and post tweets about some posts from a specific site.
For example, I can mention Math.SE. TeX.SE, Academia.SE, Physics.SE, etc. You can find twitter accounts for various sites in the list of Stack Exchange sites. However, not every site has a twitter account.
All I know about the rules, based on which the posts which are tweeted are selected, can be found here,
here and here.
Brief summary: For 8 posts a day (once in every three hours) the posts are chosen from these categories: bounties, unanswered, often viewed, interesting questions and answers, hot on meta. I am not sure whether the exact algorithm which chooses posts to be tweeted is published somewhere.
It was mentioned several times in comments - both here on meta.MO and elsewhere - that twitter accounts for Stack Exchange sites are no longer supported. (By which I mean that they are not added for new sites and no improvements and new features are added. But the twitter bots which were created in the past still exist.)
I did not find an official announcement about this, but it was mentioned by some experienced users, so I tend to believe this. (And the fact that several newer sites do not have their own twitterbots is a supporting evidence.) Mad Scientist added in a comment link to an answer on meta.SE related which mentions reasons for this decision: Why is it so difficult to find the twitter account for a Stack Exchange site? I will quote this part: "Twitter-the-company started blocking new accounts for us (probably because it thinks we're bots?) and fighting that got too annoying and time-consuming." (This is probably useful to know for anybody who considers starting some twitter account where posting of tweets based on the content of MO would be - in some way - automated.)
If you want to learn more about these twitter bots, the tag twitter on meta.SE seems to be the right place to go.
To me this seems as the "cheapest" option. There already is existing infrastructure and an algorithm which selects what to tweet. So it would probably not be too much work to add this also for MO. (However, since these twitter bots are no longer supported - one of the main reasons might be the problems mentioned above - I am not sure whether SE staff would be willing to do this.)
EDIT: Probably it's worth adding that there are some SE sites that have a twitter account which is run by community, i.e., tweets are added manually and not by bots. There are two such sites that I am aware of: Literature Stack Exchange (twitter account and announcement on meta) and Mi Yodeya (twitter account and announcement on meta).
Which questions to tweet
Several possibilities how to choose questions which should be tweeted have been mentioned.
- As I wrote above, twitterbots for other sites already have some algorithm how to choose questions.
- Hot network questions were suggested as a possibility by the OP. I'll just point out that questions with MathJax in the title are never chosen in the HNQ. Still, I guess it is possible to calculate for such questions whether question is hot or not (they are just not picked in the networkwide list.)
- Some other possibilities would be choosing questions similarly as in the newsletter (suggested in a comment).
- You can see questions listed as greatest hits and questions which are currently hot. AFAIK the choice of these questions is related to choice of question in the newsleter and the hot network questions list.
Is MO already present on twitter?
In fact, you can see that there are already many MO-related tweets out there. You can easily find some of them by searching in Google or on Twitter or using #mathoverflow. (You might also recognize some of usernames as users on this site.)
In fact, Joel David Hamkins tweeted about this meta question (and it seems that he has many tweets related to MO posts.) See also some his comments above.
I found also two twitter accounts: MathOverflow and MathOverflow Feed. However, there was no activity on those accounts since 2011. (Creation of one of them by Anton Geraschenko is mentioned in this tea discussion. I do not know about the other one.)
Users who have access to site analytics can see how much of the referrals are generated by twitter. (As I mentioned in my comment, the statistics of traffic sources for math.SE
t.co is listed among referring sites with 3.4%. This is aprroximately same level as
googleweblight.com; but lower than
mathoverflow.net with 5.9%. I do not have access to site analytics on MO.)