Short answer: Don't do it (unless you really have to).
The process you are looking for is called "upscaling". In its simplest form, every pixel in the video will be (almost) doubled or stretched and the resulting video will look like 1080p. The problem with this approach is that it doesn't look any different from a full screen option in any normal video player.
What you can do:
So if you take a video and upscale it with a very basic algorithm, and then re-encode it, the quality will be worse than before. As @boehj mentioned in the comments, you will always lose a certain amount of quality when you take an already encoded video and encode it again. Even if you upscale it and encode it with a lossless encoder, this will result in huge files – but in no noticeable difference from the full screen version.
Naturally, how a player displays a full screen version also influences your experienced quality. If the player is bad, then upscaling the video before you play it is a viable option. Of course, there are more approaches to upscaling. You can use bilinear or bicubic filters to smooth out edges and then apply some grain or edge sharpening. Our favorite codec library, FFmpeg, when compiled with the
swscale lib, can do this. It uses bilinear filtering as far as I remember.
What movie companies do
Well, most old movies are available on film. Either the stock film they used before cutting or the final film that is delivered to cinemas. As film naturally has a higher spatial resolution than any of today's HD specifications, you can just scan the negatives again and thus get a movie in HD resolution. This is what is mostly done when old movies are re-released as BluRay discs or in "digitally remastered" versions.
More recent movies are actually shot in HD. Or even more, up to 4K resolution. This means that after the movie is cut, color-corrected, etc., the final movie will be downscaled to fit distribution needs. So, iTunes might downscale it to 720p before making it available, or BluRay distributors downscale it do 1080p. This is only possible because they have the original material.
Phew. That all being said you can always download Handbrake, which relies on FFmpeg, and upscale the video using the "Size" options under "Picture Settings". There aren't that many options though.
Just try it on a video and see if you can spot any difference. If not, don't upscale, it won't pay off.