My impression is that, among many Christians, people around a table saying a prayer thanking Jesus for a meal hold hands with one another. What is the origin of this practice, and are there any reasons for it given in reliable sources?
Protestantism and Tradition
I'm coming at this answer from a Protestant perspective. Since I've seen a variety of folk in my branch of Christianity hold hands during prayer, I'll assume you've been observing my people. We've inherited from Paul a suspicion of traditional rites and practices. When we do observe some custom, we are very likely to either:
I doubt either of these will be satisfactory to you, but maybe the sociological aspects will be of interest.
No Biblical justification
Laying hands on a person (especially when they are sent off to perform some ministry) has strong support in the practice of the early church. Probably the earliest example is:
That practice has roots in the life of Jesus and in Hebrew culture. Of course, praying while putting one's hands on another isn't the same thing as joining hands. We get a hint of the idea of holding hands from this mention in one of Paul's letters:
Again, this sounds more like a handshake than a prayer circle. So the goto source for Protestants seems to be silent on this issue except that the posture of the hands can be important somehow.
There's a lot of stuff that Christians (and Protestants in particular) do for strictly practical reasons. For instance, as far as I know the only reason we have lots of guitar in our worship services is that it's an easy instrument to learn well enough to make some semblance of music to accompany singing. Churches are removing organs (the musical instrument, I mean) because it's getting harder to find decent organists. So while you will still find Christians who think God only hears music accompanied by some sort of keyboard and that guitars are from the devil, most of us are more relaxed about the whole thing. (Culture and the rejection thereof have a lot more to do with these attitudes than any sort of solid theology.)
When it comes to holding hands in prayer, I think there are several practical purposes:
No doubt there are other considerations. There's no hard and fast rules about posture in prayer, so it's not uncommon to have some participants holding hands while others place their arms over their neighbor's shoulders. (I'm often the link between the two for some reason.)
The mind-body problem
Finally, I must find a way to quote C. S. Lewis' The Screwtape Letters. It's a work of fiction from the point of view of a demon writing to an underling about how to divert a man ("the patient") from God ("the Enemy"). This is Screwtape's advice about prayer:
In other words, we must orient our bodies toward God if we are to orient our minds and souls toward Him. Christians don't have a precise requirement for how to do this, but we have traditions that we follow because they have that practical effect.