Anti-aliasing / Filtering in Ray Tracing
There is a great paper from 2006 on this topic, Filter Importance Sampling. They propose your method 2, study the properties, and come out generally in favor of it. They claim that this method gives smoother rendering results because it weights all samples that contribute to a pixel equally, thereby reducing variance in the final pixel values. This makes some sense, as it's a general maxim in Monte Carlo rendering that importance-sampling will give lower variance than weighted samples.

Method 2 also has the advantage of being slightly easier to parallelize because each pixel's computations are independent of all other pixels, while in method 1, sample results are shared across neighboring pixels (and therefore have to be synchronized/communicated somehow when pixels are parallelized across multiple processors). For the same reason, it's easier to do adaptive sampling (more samples in high-variance areas of the image) with method 2 than method 1.

In the paper, they also experimented with a Mitchell filter, sampling from abs() of the filter and then weighting each sample with either +1 or &minus;1, like @trichoplax suggested. But this ended up actually increasing the variance and being worse than method 1, so they conclude that method 2 is only usable for positive filters.

That being said, the results from this paper may not be universally applicable, and it may be somewhat scene-dependent which sampling method is better. I wrote a blog post investigating this question independently in 2014, using a synthetic "image function" rather than full rendering, and found method 1 to give more visually pleasing results due to smoothing high-contrast edges more nicely. Benedikt Bitterli also commented on that post reporting a similar issue with his renderer (excess high-frequency noise around light sources when using method 2). Beyond that, I found the main difference between the methods was the frequency of the resulting noise: method 2 gives higher-frequency, "pixel-sized" noise, while method 1 gives noise "grains" that are 2-3 pixels across, but the amplitude of noise was similar for both, so which kind of noise looks less bad is probably a matter of personal preference.