Publishing articles nowadays is terribly easy and does not cost a thing (arxiv); filtering and getting good referees however is not.
My solution for this would be a public network of papers, where everybody can publish, read and 'sign' those papers. If you agree with a paper, you put your signature under it and the worth of this paper goes up. As your 'worth' goes up your signature also gains in weight, when signing other papers. Every paper gets a comment section, where reviews can be written and errors pointed out.
If a well known professor therefore signs your work, others will catch up to it. A 'good' paper will gain in publicity quickly due to being sent around a lot. One would also need to include a system of diminishing returns, as to avoid groups signing only their own papers. Ironing out these points of abuse will be the hardest part of this system.
The specification above only consists of four to five sentences and yet I would call it more robust than the currently arbitrarily chosen journals.