On the Rømer experiments and the speed of light

So my question is, why did Rømer find the figure of 22 minutes? Am I overlooking some part of the mechanics? Or is it that his lab equipment was very imprecise because it was almost 350 years ago?

As @AdrianHoward already mentioned in his comment, the main problem was the lack of precise clocks at 1676.

If it is the latter, could I then repeat the experiment with modern equipment and find a reasonable figure for the speed of light?

Yes, you surely could repeat Rømer's experiment. And you will get much more precise time-measurements just by using a radio-clock or your NTP-synchronized computer-clock.

Has anyone done this?

The experiment was actually repeated after the invention of mechanical high-precision clocks.
Quoted from Rømer's determination of the speed of light - Later measurements:

In 1809, again making use of observations of Io, but this time with the benefit of more than a century of increasingly precise observations, the astronomer Jean Baptiste Joseph Delambre (1749–1822) reported the time for light to travel from the Sun to the Earth as 8 minutes 12 seconds. Depending on the value assumed for the astronomical unit, this yields the speed of light as just a little more than 300,000 kilometres per second.

The wikipedia article that Thomas Fritch pointed out about Rømer's determination of the speed of light is very interesting.

Quoting from that article:

From the Earth, it is not possible to view both the immersion and the emergence for the same eclipse of Io, because one or the other will be hidden (occulted) by Jupiter itself. At the point of opposition (point H in the diagram below), both the immersion and the emergence would be hidden by Jupiter.

For about four months after the opposition of Jupiter (from L to K in the diagram below), it is possible to view emergences of Io from its eclipses, while for about four months before the opposition (from F to G), it is possible to view immersions of Io into Jupiter's shadow. For about five or six months of the year, around the point of conjunction, it is impossible to observe the eclipses of Io at all because Jupiter is too close (in the sky) to the sun. Even during the periods before and after opposition, not all of the eclipses of Io can be observed from a given location on the Earth's surface: some eclipses will occur during the daytime for a given location, while other eclipses will occur while Jupiter is below the horizon (hidden by the Earth itself).

The usual depiction of Rømer determination is to describe observations precisely at the ideal points: one where the Jupiter-Earth distance is at its largest, another when the Jupiter-Earth distance is at its shortest.

It is clear that the data that Rømer worked with were not that ideal case at all. Rather, he would have a set of observations during a period of increasing distance between Earth and Jupiter that showed a trend to progressive delay, and during decreasing Jupiter-Earth distancea a set of observations with a trend to decreasing delay.

Also, the orbit of Io is affected by the the other Jupiter satellites. The effect is smaller than the Rømer-effect, but not zero.

So rather than having direct measurements Rømer had to arrive at a result by extrapolating trend lines.

Oversimplification

It think it's safe to say that the usual presentation of Rømer's case for a finite speed of light is oversimplified.

The oversimplication is understandable, but it does have a big disadvantage: the error (11 minutes versus 8 minutes) becomes inexplicable.

I find it interesting to see Newton's attitude to the case for a finite speed of light. In the 1704 book Opticks Newton wrote:

Light is propagated from luminous Bodies in time and spends about seven or eight minutes of an hour in passing from the Sun to the Earth. This was observed first by Romer, and then by others, by means of the Eclipses of the Satellites of Jupiter.

Newton endorses the case of a finite speed of light. Presumably Newton had access to better data, allowing him to give a more accurate estimatation. Still, Newton didn't bother to try and arrive a value down to seconds. I assume that is because with the available data the 'seven or eight minutes' was the best estimate possible. I infer that even for Newton it wasn't about determining an actual value for the speed of light, it was about whether the speed of light is finite or infinite.

Also worth noting: Roemer characterized the quality of each of his measurements in his journal, based on the atmospheric seeing and other factors. IIRC, an article on this topic appeared in the American Journal of Physics where all they did was apply Roemer's own methodology but selected only those measurements that Roemer himself classified as good quality. This produced a value that was astonishingly accurate. Unfortunately I can't seem to find the article in google, only people talking about it.