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So what would be an example of an unfair or wrong weighting? Or, put another way, what's wrong with New Hampshire that would be improved by taking on some of the characteristics of states at the other end of the ratings list?

Their weighting of local, state and federal government employment and spending. From the report is this quote

Analysis
Alaska’s big problem is fiscal policy. Over a quarter of the state’s workforce is employed by state or local government, and that figure does not include federal employees. Alaska has the highest debt and government spending to personal income ratios in the country. However, Alaska does extremely well on personal freedom, scoring fifth on our ranking. Reasons for its score include fully legalized possession of small amounts of marijuana (accomplished through a court ruling), the least-restrictive gun laws in the country, strong asset-forfeiture protections, recognition of same-sex domestic partnerships, and possibly the best homeschooling laws in the country. On economic regulation, Alaska does poorly on labor law, occupational licensing, and eminent-domain reform but relatively well on health-insurance regulation.

OK, that sounds like you think Alaska should rank higher than it does since there is too much emphasis in the ratings on the percentage of people employed by the government. I presume you don't think other states should emulate that, but that the measure unfairly downgrades Alaska due to its small population and unique circumstances.

All rankings suffer from trying to map a multidimensional set of characteristics into a single ordered list, it would be interesting to see how the Freepers would rate the states.