As a wet lab scientist, not a true bioinformaticist (by any stretch), I think one of the largest problems is the use of so called "in house scripts" for analysis, the details of which are not disclosed in the methods section.
I couldn't do biochemistry and write in the methods section that "the reactions were run using in house buffers" nor could I publish an novel experiment without a thorough enough methods section to enable someone to attempt to repeat the work. The field has long agreed on what needs to be disclosed for bench science, but has not yet come to a consensus with genomics. In many of the genomics experiments with which I am familiar, the analysis is as crucial if not more so to the conclusions of many papers than then preceding bench work, yet the methods sections discloses relatively little about how the data were analyzed.
Perhaps the problem is due in part to the fact that many of the reviewers in the fields of biochemistry and molecular biology weren't trained in genomics and couldn't adequately review the genomics methods even if more detail was provided. In other words, it might be more the fault of the molecular biology establishment that the current generation of genomics and bioinformatics researchers.
This limited disclosure of methods, has, I think, contributed to the code duplication that exists for common analysis such as mapping ChIP-seq reads to reference points, binning the data, clustering, etc. I realize biology and the nature of experiments dictates to some degree the type of analysis that must be carried out, and some customization is required, but it would be a great benefit to the field if bench scientists began to coalesce around some of the tools freely and publicly available so that the tools and parameters used in an analysis could be accurately and concisely cited.
Lastly, think about how much money the NIH has spent paying graduate students and post docs to develop the same/similar tools for analyzing data from something like a ChIP-seq experiment, all in a very difficult funding environment. With countless labs having countless variations of similar tools, some of which may not be readily available, curated, or supported after students leave, I don't see the problem improving soon.