What is the origin of holding hands when praying around a circle before a meal?

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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:

create a post hoc justification, or
protest that the custom is not binding (and we can stop at any time).

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:

These they set before the apostles, and they prayed and laid their hands on them.—Acts 6:6 (ESV)

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:

and when James and Cephas and John, who seemed to be pillars, perceived the grace that was given to me, they gave the right hand of fellowship to Barnabas and me, that we should go to the Gentiles and they to the circumcised.—Galatians 2:9 (ESV)

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.

Practicalities

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:

It symbolizes that we are all equal before God and creates a sense of unity. (See the Galatians passage above.)
It shows who is participating in the prayer. (This can be helpful when praying at a restaurant to avoid the awkward situation of a server interrupting a prayer.)
When praying with small children, it ensures small hands are not making mischief while eyes are (theoretically) closed.
It allows for silent communication (a squeeze of the hand) during the prayer.

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:

The best thing, where it is possible, is to keep the patient from the serious intention of praying altogether. When the patient is an adult recently re-converted to the Enemy's party, like your man, this is best done by encouraging him to remember, or to think he remembers, the parrot-like nature of his prayers in childhood. In reaction against that, he may be persuaded to aim at something entirely spontaneous, inward, informal, and unregularised; and what this will actually mean to a beginner will be an effort to produce in himself a vaguely devotional mood in which real concentration of will and intelligence have no part. One of their poets, Coleridge, has recorded that he did not pray "with moving lips and bended knees" but merely "composed his spirit to love" and indulged "a sense of supplication". That is exactly the sort of prayer we want; and since it bears a superficial resemblance to the prayer of silence as practised by those who are very far advanced in the Enemy's service, clever and lazy patients can be taken in by it for quite a long time. At the very least, they can be persuaded that the bodily position makes no difference to their prayers; for they constantly forget, what you must always remember, that they are animals and that whatever their bodies do affects their souls. It is funny how mortals always picture us as putting things into their minds: in reality our best work is done by keeping things out.—The Screwtape Letters, Chapter 4.

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.