The low R-squared is presumably because you use daily data. If t is significant (or the F-statistic for the whole regression, though F will almost always be significant if at least one of the t-stats is), then it means that beta is successfully explaining something… it’s just that there’s a lot of “other factors” that explain day-to-day differences from the index. If the stock is illiquid for some reason, it wouldn’t be too surprising if day-to-day mismatches in supply and demand push the price around a lot on a daily basis and dilute the effects of systemic factors.
If you switch to a weekly or monthly return series, a lot of the daily noise will likely wash out, and that should improve your R^2. Your beta could change, but it’s most likely to remain around the same.
You generally want to measure beta across a full market cycle. The way beta behaves in an up market and a down market can be different, and so using a full market cycle tends to reduce the effects of this by averaging it out over what can be expected long term. If you only regress numbers from an up cycle or a down cycle, your estimates are likely to be biased in whatever way beta behaves during that part of the cycle. Remember that WACC and CAPM type required returns are intended for long-term estimations. So basing your estimate on 3 years is really problematic (though if the company has been listed only 3 years, maybe you have little choice - in that case, the option Mr. Smart mentioned is clearly a better option).
The low beta suggests that the stock is not very correlated or responsive to the local market index. That could be because it’s an international company and correlated to something else, or it could simply be that there’s a more important thing that’s driving the stock price (say, the international price of a key input like oil or something). Or it could be that it’s a utility or staple goods company, or that it’s a state-owned/dominated enterprise that has a state-mandated profit margin, so that the value of this stock is expected to be supported in some way by state subsidy.
If you abandon this method of computing beta, based on the idea that the market doesn’t do very much to affect the price, you might go ahead and use the average historical return for the stock. But my guess is that it won’t be that different from the CAPM number, because beta is small and R^2 is high, so you might not improve things much. You could also try to figure out what the other driving force behind profits are and see if you can include that into an expanatory variable in your model (though this would then require a long-term prediction for that variable in order to turn it into an estimate of the return from that factor).
The bottom-up beta idea isn’t necessarily bad, but it’s harder to do than you’d expect because it requires you to find comparable companies and then calculate the unlevered beta for those companies, and then assume that your company is effectively the same as they are (other than the capital structure). In many places, the only comparables are operating in other countries, which is a different enough context that the betas may be markedly different, and estimated off of different market indexes in the first place. Even if the comparables are in the same country and industry (which usually requires that there be a large and diverse economy to support enough listed companies in the space), you still run into the problem that to calculate the beta for those companies still requires regressing historical data before unlevering them, so the errors that potentially come from the levered beta estimate in the beginning don’t actually disappear, they just get averaged together with everyone else’s errors.