With emphasis on "successful". This objective is new and I am currently working on it full-time.
This rest of the list is really, really old. Regard it as obsolete (not that I have achieved all those objectives, of course …)
The Research Center Jülich is in the process of acquiring a new supercomputer, a cluster of IBM Regatta nodes, as a replacement for the aging Cray T3Es. Of course, the parallel environment differs drastically. Porting tools like a SHMEM-emulation library are supplied. Try them, test them, learn MPI (recommended by IBM), learn the different tools (performance measurement, debugging, etc.), port old programs, do speed tests.
Use parallel Hoshen-Kopelman for investigating various critical exponents, specifically corrections-to-scaling in two dimensions (because in two dimensions we have the highest chance to get new insight). For two-dimensional percolation, the domain decomposition into strips parallel to the HK-hyperplane seems superior over the decomposition into perpendicular strips (the opposite is true for higher dimensions). This should be checked (by simply trying both).
Using the Newman-Ziff algorithm, a lot of other interesting things can be studied efficiently, especially critical exponents of the kind (p-p_c)^x.
Another important project would be to study finite-size effects thoroughly, especially in conjunction with varying boundary conditions. This could yield high precision results with simulating very small lattices.
How do social hierarchies form? What do they look like? Can we even explain dynamic effects (like revolutions)?
One important model for studying social hierarchies was presented by Bonabeau et al. Old results seem to be wrong, Sousa and Stauffer found new results. A deeper look could be needed. Other models could be interesting, too.