One simple method, based on the principles of mean-variance optimization, is to set the weights proportional to the product of the inverse of the covariance matrix and a vector of standard deviations. This implicitly assumes that the normalized expected return of each stock is equal. If you wish, you can take only the top 5 weights and set the others to zero. The actual problem you face, of selecting just 5 stocks, can be solved rigorously with an optimizer, but since it is not a quadratic program, may be difficult to solve.
Update
A more sophisticated but very interesting additional possibility is to find the "Maximum Diversification Portfolio (MDP)", as defined in Toward Maximum Diversification (free version, hat tip vonjd). The MDP is defined as the portfolio that maximizes the Diversification Ratio (DR), which in turn is defined as the ratio of the portfolio’s weighted average volatility to its overall volatility. A follow-up paper investigates the properties of this portfolio. From the paper:
This measure [DR] embodies the very nature of diversification whereby the volatility of a long-only portfolio of assets is less than or equal to the weighted sum of the assets' volatilities. As such, the DR of a long-only portfolio is greater than or equal to one, and equals unity for a single asset portfolio. Consider for example an equal-weighted portfolio of two independent assets with the same volatility: its DR is equal to $\sqrt{2}$, and to $\sqrt{N}$ for $N$ independent assets.
$DR(\bf{w})=\frac {\sum_i{\it{w}_i\sigma_i}} {\sigma(\bf{w})}$
