There are a few premises that must be established to make a case for BI.
1. We are approaching a post-scarcity economy.
2. In a post-scarcity economy, many people will lose the ability to support themselves\their families, as whole industries are automated.
3. In the case of massive unemployment, consumption will drop dramatically.
4. Drastic drops in consumption will devastate the economy. This will be bad for everyone. Less wealth, less progress, less investment.
5. Therefore, to maintain economic growth/stability in a post-scarcity economy, we should provide a basic income.
That's the best argument I can formulate. It really has nothing to do with having fun and not being forced to do menial work. Those are fringe benefits.