People have chosen different ways to pick factors. Chen, Roll and Ross are a classic example of attempts to find reasonable ICAPM factors. Fama-French factors are often explained as correlated with underlying ICAPM factors. Other researchers have chosen to look for factors without assuming outwardly observable exposures by analyzing returns using factor analysis or principal components.
Explanations of Fama-French
I don't know if there are "commonly accepted" explanations for the Fama French factors. This is not to say that there aren't explanations, there are a lot of them. Just no one seems to agree on which ones are "best". Candidates include both consumption based explanations AND production based models, each of which relates Fama-French factors to underlying economic variables. Cochrane has a nice summary of the performance of both in pricing Fama-French portfolios here.
Additional factors commonly used
Fama and French are now talking about a five-factor asset pricing model which also includes profitability and investment, similar to Novy-Marx's paper and work by Chen, Novy-Marx and Zhang which more directly relates to production asset pricing. Momentum is also a consistent candidate, although it's a real pain to put any kind of explanation to. There are tons of others. This paper lists many of them, as well as calling the significance of some of them into question.