I have been playing the vibraphone for about 8 years. I am left handed and i find it easier to play the complex melody with my weaker right hand and the chords with my left. It could be that the stronger hand has less trouble being the more 'independent' of the two hands when playing a musical instrument. Once playing, the stronger hand can be forgotten about and keep playing whilst what the weaker hand has to do can be focused on by the musician. 

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Left-handed people who wish to learn a musical instrument will have a hard time of it.  Most musical instruments are asymmetric and favour right-handers (even the humble descant recorder has one hole at the end offset to favour the right-handed player and, I believe, the internal structure of violins is designed to enhance the various pitches of the strings so that it may not sound right if you simply switch the strings round to play left-handed).

As for a right-handed guitar; it may appear that the left hand has the most difficult and complex job to do, but that really only applies to rock guitar.  Watch a good classical, flamenco, folk or jazz guitarist and you will realise that the right hand has a more important rôle.  What it comes down to is that you play a right-handed guitar with your right hand – the left hand simply chooses which notes you play.  This is also apparent with other stringed instruments.

I believe that a similar bias applies to other instruments such as keyboards and wind instruments where the actions of the hands are superficially equivalent; in a more subtle but real sense the playing is ‘led’ by the right hand.  Consider that Paul McCartney is a superb bass player but only an average pianist!

I am a left-hander who learnt to play a right handed guitar for precisely the reasons outlined by Tony Baker in his question and have regretted the decision ever since – as a consequence I am merely a passable guitarist and not the rock star I could have been!  I cannot urge strongly enough that budding left-handed guitarists should buy a left-handed guitar – or take up drums.

I am neither a guitarist nor violinist, but I believe that the movements required by the non-dominant hand are primarily in the fingers. And while you may be right-handed, you aren't really "right-fingered." To illustrate this, you can type about as fast with your left hand as you can with your right. So your left hand is probably about as suited to stop the strings on the fretboard as your right hand is.

On the other hand (quite literally), the motions that your dominant hand makes while playing a stringed instrument do greatly benefit from the additional dexterity in the hand and wrist.

All musicians rely on muscle memory as part of their skill development hence the repetitive exercises as well as repeated practice and rehearsal of pieces. When playing instruments like a guitar which can be strung for right or left-handed play it makes more sense to string it so that the weak hand is on the fretboard.

During the initial learning stages the left hand is playing chords or fretting a note while the right hand is strumming or picking strings to sound the note. As the right hand cannot be watched actively with ease the player tends to watch where they are putting their left fingers to improve accuracy while relying on the greater precision of the right hand to hit or pick strings without looking.

Additionally when playing a guitar the left fingers have the least to do - press on a string or number of strings, 'bend' a string or move to produce vibrato and generally move slower while the right fingers have to hit or pick the strings with varying  pressure, speed and rhythym or pick individual strings in complex patterns. The guitiar is one of the few instruments where the strong hand is responsible for both the sounding of the notes and the rhythm in contrast with as an example the piano where generally the left hand handles rhythm while the right produces the melody, although there are exceptions.

For guitar players once a piece is learned it can be played subconsciously and it is easier to introduce variance by conscious control of the strong hand.

The nearest accessible analogy is in First Person Shooter (FPS) game play on a PC where the direction of movement i.e. the leg movements, are controlled by A, D, W and S or arrow keys and the mouse is used to control the finer and more precise hand movements. In this case right hands prefer to use their left hand on the keyboard and the option exists for left handers to use their right hand to manage the keys and control the mouse with their left.

I have been playing fretted and non-fretted string instruments for a long time, as well as causally teaching friends who wish to learn. I find that beginners often make the assumption that the left-hand movement (when using a conventional instrument) will be harder to learn than the right. It is my experience that, deceptively, the right hand is doing equally, if not more 'complicated and extended fine-motor movements'.

For example, with your dominant hand, see how long you can tap out an even rhythm with the following finger pattern. 2-1-3-1-4-1-3-1, where 1= thumb, 2 =index finger, etc. A tempo of eight taps (one 'cycle') every second, would not be unreasonable for a competent guitarist, with an emphasis on accuracy and an even beat. Now try alternating each 'cycle' loud and soft; an aspect that the right hand is also responsible for. This finger pattern could be used with a static chord shape in the left hand. When learning, the fingers on the left hand can be placed fairly clumsily (more so with fretted instruments, I would argue) and still achieve the desired result. This is similar to a 'finger-picking' technique.

A strumming movement is also deceptively tricky to master, and more akin to the bowing technique used on classical string instruments. I can only liken the learning of correct bowing technique to that of learning to hold a pen 'correctly'. Very precise, minute pressures are applied almost imperceptibly (and, once the technique is developed, unconsciously) by each digit. Musicians using bowed instruments also often use chord shapes in the left hand, in order to minimise necessary movement, and hence increase playing speed.

For fret-less instrument, one also needs to learn the position of the notes, either by ear training, or temporary marks. This is in addition to the left hand posture. It is perhaps this combination that has given the violin its reputation for not giving producing the most soothing tones when learning.

Presuming that the right hand takes the more complicated task, the only instrument that I can think of where this may not hold is the accordion

I believe the theories of brain lateralisation (left-brain / right-brain function association) are generally over-simplified or misunderstood. However, there may be an additional societal reason. The majority of people are right handed, while at the time of the development of these instruments, left-handedness was associated with disruptive traits in many cultures.