Ah, this is something I know quite a lot about, so hopefully I can provide a good answer, although its rather complicated and messy.

The short answer is that it meant 'the common law', the body of legal precedent which had built up over the centuries. The slightly longer answer is that it depends who was discussing it and for what purpose they were invoking it.

While people across a broad range of political and constitutional views agreed on the existence of an ancient constitution of some sort, its exact nature was contested, and some historians have argued that the turmoil of the mid 17th century was about defining this 'ancient constitution'. If you want a book which will explain this better than I can, I would recommend Glenn Burgess's The Politics of the Ancient Constitution, which is a bit old now but still stands up very well in my opinion.

To actually attempt to answer your question of what, historically, is being referred to, the most common answer is the Saxon kingdoms. Early modern English writers were very interested in the Saxons, who, many of them argued, represented the 'true' English constitution which was later subverted by Norman absolutism.

Exactly how far this argument was taken often depended on who was making it. The idea of a tyrannical 'Norman yoke' was a popular one amongst Civil War era radicals such as the Levellers, but a wide range of writers invoked the idea of Saxon precedents, not always necessarily in such radical ways.

There were several aspects to the 'Saxon constitution' which appealed to early modern writers, particularly those opposed to Charles I. I will caution that I can't say whether any of this is a true reflection of Saxon history, as that is not something I know very much at all about, aside from through the prism of how early modern writers discussed it, but the point is that this is what contemporaries believed, or professed to believe for rhetorical reasons, the Saxon constitutions involved.

The Saxon kingdoms, they argued, were not absolute monarchies in the continental style, but mixed monarchies where the king ruled in concert with a proto-parliament (referred to as a Witenaġemot, though the historicity of this term isn't clear). The appeal of such arguments for Parliamentarians seeking to restrain the king is obvious.

Saxon precedent was often also invoked in religious discussions. One of the most important aspects of the English Civil Wars is that they represented a crisis of the relationship between religious and civil power. In many ways the Civil Wars were a contest between different models of church and state, such as the Episcopal model favoured by the Royalists, the Presbyterianism of the Covenanters, and Cromwell's Erastianism.

Saxon councils, it was argued, were composed of both lay rulers and clergy, who cooperated without infringing on each other's respective spheres. In a time when the question of the relationship of church and state was one of great concern, precedents like this which could be invoked as a potential solution had obvious value. This was also a time of rampant fear and hatred towards Catholics, and suspected crypto-Catholics, and in this context the Saxon church was invoked as being an example of how 'true' churches functioned before their corruption by the forces of popery.

Saxon precedents were by no means the only historical precedents which writers looked to in this period, with some seeking to emulate, for example, the Hebraic republic, the Roman republic or the patristic church, but they are probably the most commonly associated with the 'ancient constitution'.

A similar process occurred at the same time in Scotland, where 'Covenanters' sought to defend their Presbyterian church and mixed-monarchy constitution which had, in essence, emerged relatively recently in the Scottish Reformation of the second half of the 16th century. Some invoked the concept of Pictish precedents, others a continuity with the Christian 'barbarian' kingdoms of late antiquity. One of the leading Scottish religious and political figures of this period, George Buchanan, was as much a historian as he was anything else, and used his histories as defences of the contemporary Scottish constitution.