Instead of using more quantitative methods of measuring power such as military effectiveness or economic or industrial power, I would argue that a more meaningful and perhaps more enlightening way of viewing this issue is through the eyes of contemporaries.

A few things are quite clear even to the average student. Italy, since its unification, had a history of playing second-rate power compared to the European heavyweights. Before the Great War it was considered a notable power by the "Great Powers", and even in this period it was perhaps a borderline "Great Power". It was certainly strong enough to be included in the pre-war Alliance game, though of all the countries that were, it was the weakest.

Its performance during the war justified this view. Despite being on the winning side, the Italian military had failed to display any demonstrable success or, in the eyes of the British and the French, any real contribution. As a result, in the resulting peace, Italian demands (including previously-made promises by the Allies), were essentially ignored and for the most part left unsatisfied. This, combined with post-war difficulty, did contribute to the rise of the fascist state.

It's important to have this understanding of Italy; that, for a long time it was viewed as the worst of the Great Powers if one at all. Even during the interwar period, when the USSR had essentially exited European diplomacy and Germany was disarmed and democratic, these two latent powers remained the focus of British and French diplomacy, and a significant amount of dealings and agreements with the Italians were made with this focus in mind.

When Hitler came to power in Germany in 1933, this trend picked up even greater traction. As a ideological successor of sorts of Mussolini, Hitler obviously was at risk of quickly taking Italy into its sphere (though this was not how Mussolini would have seen any sort of alliance). This loomed large on the minds of British and French diplomats and they made a concerted effort to keep Italy from consolidating into a bloc with Germany. What some people might see now as a foregone conclusion, the Rome-Berlin Axis, was in 1935 not at all a sure thing. The British and French had very reasonable hopes to keep Mussolini's Italy friendly and at arm's length from Germany. The Rome-Berlin relationship was not completely smooth and had numerous bumps, and Mussolini was never a great fan nor friend of Hitler.

The British and French then found themselves in a diplomatic quandary. Mussolini was wildly ambitious and hoped to fashion a new Mediterranean sphere of influence for Italy, something that would come at the cost of the sovereignty of numerous other states, not to mention complete destabilization of delicate interwar Europe. So the Allies had to keep Mussolini reigned in. This was much easier, however, before Hitler's shadow rose over Europe. The British and French then had to simultaneously appease Italy while reigning it in, walking a delicate line.

The events of the Ethiopian war ended in disaster, when the British and French were exposed as trying to appease Italy despite publicly professing a dedication to sovereignty and the principles of the League of Nations. Italy would leave the League in the aftermath and relations would never be good again between the Italians and the British and the French. This was a crucial shift.

During the re-militarization of the Rhineland, which happened shortly after Ethiopia, Italy kept a pointed silence, refusing to criticize Germany or take any sort of action despite the violation of Locarno, where Italy had promised to serve as a guarantor of European peace. Locarno was considered a centerpiece of the "stable" European situation from 1925 to 1930, and in 1936 was still mentioned as a bulwark for peace. The events of the re-militarization and Italy's icy reaction meant the death of Locarno to contemporaries (though historians might say it was already dead).

All this summary of Italy's interwar history is simply to show how important or unimportant it was considered by contemporaries. And its inclusion in Versailles and Locarno, though of a lesser nature than its stronger European counterparts, revealed that Italy was "part of the club". The British and French certainly thought so enough to make efforts to court the Italians, as well as enough to consider it a blow to European stability when it left the League and abandoned Locarno.

The Italian military (including its navy), which later proved to be much less effective than imagined, factored largely in at least the British minds (and maybe the French). Rearmament efforts were driven not just by the German rearming but also by the Italian threat to the British Mediterranean sphere. Some British leaders in fact pessimistically evaluated that the Italian military could impose naval and air superiority over the British in the Mediterranean.

I certainly think that, taking the way Italy was treated by the great powers of the time, that it was indeed a Great Power by even a conservative definition of the term. However, it was undeniably a very weak one in relation to the others.