I mean, technically this could just make them a secondary Germany. Famous round the world for their military successes in a war that was doomed to fail once Germany invaded Russia and Japan attacked Pearl Harbour. It would change the cultural perception of the Italians a lot, and they'd be villains/anti-heros in more epic war movies. People would geek out about Italian uniforms, medals and tanks in the same way they do about German memorabilia. Mussolini would be seen as less of a buffoon too. People would talk about WWII and say "we'd all be speaking Italian" about certain battles and make wild AH scenarios where Italian troops march through the streets of Washington DC.
They don't even need to get close to winning in reality, just enough "falling with style" defeats to convince people that Italians are totally badass.
What the nature of those successes would be and what they'd need to do to prepare for them is another matter though. Totally wiping the floor with Greece and Yugoslavia without German help would be a start, though frankly while Greece gets a lot of kudos for holding back the Italians in OTL if the Italians had actually won it would probably have been seen as just a case of the weak being beaten by a stronger nation. That would have the prestige of the German invasion of France, which shocked the world. Maybe the Italians could play a far larger role in France, but I think success in Africa and perhaps at sea would have to be where they win their spurs.
Being good enough at tank warfare and having good enough generals to replicate the success of Rommel and the Panzercorp without Rommel and the Panzercorp or perhaps in a way that make Rommel and the Panzercorp seem like minor players in the North Africa campaign, perhaps even getting a little further. Breaking through Egypt and reaching the very doors of Jerusalem seems a suitably epic point for a highpoint of military success, whereupon they will suffer a defeat from overextension and overconfidence. A Stalingard style battle in the Middle East might get the gritty war movies and generals memoirs rolling off the press once the war is over.
How far south they could reach into Africa might be another matter. The deserts are sort of in the way, so I'd imagine they'd need significant naval success to launch invasions further down.
Maybe a stronger Italian navy could fight more naval battles, maybe out in the Atlantic. They'd still get trashed eventually, but the prestige they'd have from a few bigger engagements would do more to improve the notion Italy "did great".
You'd need some quite big PoD's in the military and industrial pre-war history of Italy to get any of this, in particular they need better hardware and they need a component officers, generals and military doctrines to rival that of Germany.