What brain skill does Nonogram (Picross) develop?
I could not find any examples in the literature where the game has been explicitly studied for brain training purposes. I should also confess that I don't know a lot about the specific game. However, based on general knowledge of the skill acquisition literature, I would predict the following:
Playing the game should develop your skills in Nonogram Picross. If you prefer the brain-terminology, you could say that playing Nomogram Picross develops the "brain skill" of playing Nonogram Picross. In some senses any human skill involves the brain.
I don't imagine that playing the game would improve your cognitive capacity in any general way. However, it might make you better at playing other similar games. Skill acquisition tends to be fairly domain specific.
If you are looking for advice, play the game if you get intrinsic enjoyment from it. If you want to improve your functioning in some other domain of life, then practice and learn skills more relevant to that other domain of life.
You may find it helpful to read another question on brain training on this site for more links to literature on evaluations of brain training.
Psychological video game research began in earnest in the early 1980s "as a reaction to Surgeon General C. Everett Koop’s provoking statements about the adverse effects of video games." (Mulligan, 2008) From then on, video game research tried to understand wether playing violent video games increased aggressive behavior. For 20 years, violence was the only focus for psychological science, and understanding this became even more urgent in the public mind, when it became known after the Columbine High School massacre, that the assailants had "trained" for their attack by playing Doom.
In 2003 a paper by Green and Bavelier published in Nature had a revolutionary impact on computer game research. Green an Bavelier found that playing action video games increases visual attention capacities. Since their article, a multitude of papers has been published, documenting research into the effect of computer game play on cognitive abilities.
A research paper title illustrating this new trend in computer game research is "Mom, let me play more computer games: they improve my mental rotation skills" (Cherney, 2008). It has been suggested, that playing action computer games would benefit women (Spence, Yu, Feng & Marshman, 2009) or older people (Kato, 2010) to increase their performance in certain cognitive skills and, for women, counter the cultural gender bias in learning these skills (which favors men), or, for older people, to counter the effects of aging. Spence and Feng (2010) emphasize the positive effect computer games can have on those skills (see the list below) that are important for success in science, technology, engineering, and mathematics:
A summary of the research is given in this meta analysis by Spence and Feng (2010). They give a table that details which games train which skills. I'll not give the complete table here, but in summary:
Action Games train:
Maze and Puzzle Games train:
I don't know Picross, but if it is a puzzle game (as your link suggests), it should increase mostly the three cognitive abilities mentioned above, and, to a lesser degree, sensory detection, attention (specifically capture and divided attention), and working memory. The article by Spence and Feng (2010) provides a good overview and is a great starting point for further research.
Edit re Jeromy's comments below (post doi to the interface at http://www.doi.org to find the articles):
There have been a few studies showing increased performance in a non-game task after training with a computer game designed specifically for this ability (so called "serious games"). Examples are games improving treatment adherence in medical patients, surgical skill in doctors etc. (overview in Kato, 2010, see above; or doi:10.1542/peds.2007-3134), training multitasking in a supermarket with stroke patients (doi:10.5014/ajot.63.5.535). But you have asked for transfer from a generic (action or puzzle) computer game to a non-related real-world task.
(doi:10.1080/001401300184378) have shown that the treatment group (trained with a computer game) was "less affected by concurrently performed interference tasks", which the authors attribute to an increased ability to perform under high cognitive load. (doi:10.1111/j.1467-8535.2006.00593.x) found that "gameplaying was more effective than drills in promoting maths performance, and cooperative [vs competitive] gameplaying was most effective for promoting positive maths attitudes".
As always, I have methodological objections agains both studies, and I believe you will have, too. (doi:10.3389/fpsyg.2011.00226) gives an overview of transfer studies and their results -- and methodological shortcomings.