The thing is, there will most prolly not be a good relationship between Bitcoin and governments, and that's basically just cause Bitcoin is decentralized and governments like to work with only centralized services that they can control either complete or to a certain extent, it's quite clear to governments of today that it's impossible for them to get a foothold on Bitcoin, thus they will rather sanction/ban/regulate it than adopt it.

Having said that, as far as a country's economy is concerned, I think there are very few ways Bitcoin can be of help, Bitcoin is money and you'll most likely have to have money in that to get Bitcoin or offer one service or the other to get it, it is not distributed for free, and even if many citizens of a country hold Bitcoin, I don't see how it can positively affect the economy. It's the duty of the government to provide what is necessary in order to improve their economic situation, if the government does such, then Bitcoin can only be an addition.