Gowers wants to do a Polymath project on the origin of life. This problem has been around for a long time now -- Schrodinger wrote a not-very-good book about it in his decadent Irish phase -- and part of the trouble with it is that it's hard to make precise. As Gowers explains, some statistical physicists have been trying to model life using sandpiles etc., with the working definition that "self-organized criticality" -- the persistence of structure on various length scales, hence some manner of fractal structure -- defines life. This is an unsatisfactory approach because (1) having order on various length scales does not imply fractality, which requires that the order must be the same, which isn't necessarily the case with life unless you buy some version of the Gaia hypothesis; (2) it presupposes a property of early organisms -- that they formed multilevel ecosystems -- that isn't a priori obvious. There are similar issues with Conway's game of life and some of the computer-science approaches that involve looking for self-generated Turing machines (!) in simple games.
On the whole I'm unconvinced that the problem is a good one for a mathematician (or physicist) at present, although -- like all semi-masturbatory projects -- it might stimulate developments in pure mathematics. The hard part is not finding an algorithm that generates "life" from the "primeval soup" for some definition of life and primeval soup; it is finding one that gets the gross features of real primitive life right. The existence proof is not the issue. One needs to think harder about testability than Gowers seems to want to: there are probably a gazillion different automata that give structures that are lifelike, but are any of them relevant to what happened in the primeval soup, and how would we know?
It's not obvious, either, that there is a math-problem/toy-model/"universal" aspect to biogenesis. As I understand the rough story, it goes something like this: given a primal soup with appropriate ingredients and dreadful weather, you can form (with some finite probability) a very primitive self-replicating strand of some (probably zipping) molecule. This is more or less what the Miller-Urey experiment suggests. The gaps in this story are essentially about rates -- would the amino acids survive for long enough to run into each other, what's the lifetime of an RNA strand, etc. -- and have no math content. The next question is how an RNA strand turns into a self-replicating proto-cell. Once again, this is mostly a question of rates; it's clear that a cell is more stable than a strand, once formed. There's a whiff of universality here -- for some simplified model of e.g. 3 cell ingredients, a spherical cell ingesting amino acid at a constant rate r etc., you can ask what the conditions are for life to be stable for long enough that cells can diffuse out -- they're unlikely to be motile at first -- into new environments before they suffocate in their own waste. In principle this might give you constraints on the origin of life. However, it seems exceedingly unlikely that a model of this kind would make testable nontrivial predictions. To get any further one would have to separate the essential and accidental features of the most primitive cells, and I'm not sure we're in a position to do that. The simplest organisms still existing today are wildly unrepresentative being the ones that survived.