Like all good questions, we must set the stage before we can answer it.

So shortly after the Risorgimento in 1861-2 Italy had been unified, but was still missing significant portions of land that we would associate with a modern Italy. Lazio, the province in which Rome resides was still held by the Pope, and the Veneto was still held by the Austro-Hungarian empire. Besides there were smaller areas of various levels of contested quality between the Italian and Austrian governments like Trent.

One of the ways that the Italian government sought to make up that ground in the Veneto was to join the Prussian-led North German side in the Austro-Prussian war of 1866. Italy's performance in the war was abysmal and though the initial treaty removed the Veneto from Hapsburg grip, it was actually given to France in the treaty. Though it would end up in Italian hands eventually, the long and short of it is that Italy was more or less the laughing stock of Europe for failing to secure the lands that she claimed to hold sovereignty over.

So when Benedetto Cairoli went to Berlin for the famous Berlin congress which sought to redraw the Balkans, he claimed that Italy would "go to Berlin with her hands free and wishes to leave it with her hands clean." An attitude that would not make expansionists and colonialists happy. After all, to them the decline of Ottoman authority should mean and increase in Italian territory.

Let's fast forward to the 1870s. The rising influence of Sicilian radical Fancesco Crispi began to mark a distinct colonial ambition in liberal Italy. Almost as soon as the romantic visions of the Risorgimento had been more or less realized, vague recollections about Italy's ancient dominance of the Mediterranean began to take hold. Italy's claim to Tunisia was not entirely fabricated, in 1868 a treaty had been signed with the Bey of Tunis to secure Italian fishing and tobacco-growing rights. Not only that, but there was a large contingent of Sicilian migrants who had settled Tunisia. However late-imperialism was, at least to the claim of most governments and explorers, for the benefit of commercial enterprises, and the flat fact was that France had significantly more invested in Tunis than Italian firms did.

Cairoli assumed that the Italian and French interests would more or less both coexist in Tunisia without conflict. This led to some naive assumptions about the French colonial ambitions in North Africa. When Crispi and his party sought to assert claims in North Africa, the Left was concerned that the ensuing tensions with the French would jeopardize their ability to play the French off the Austrians to help get some of the contested territory like Trent. Italy's claim to Tunisia would collapse entirely once the French declared "that she would never allow a country but herself to establish a protectorate in the Regency." When Italy failed to challenge that claim, the last vestiges of Italy's claim, on a diplomatic scale, to a Tunisian protectorate died. Now this is not to say that Italian expansionists were not interested attempting to maintain that claim and when Crispi became prime minister in 1893 there had not yet been a formal Italian recognition of French Tunisia. Italy would not formally recognize a french Protectorate in Tunisia until after Crispi's failed war in Ethiopia cost him the position of Prime Minister for a second time in 1896.

TLDR: Italian, particularly Sicilian, contact with Tunisia and some commercial ties gave some expansionists a real interest in acquiring Tunisia, largely to verify Italy as a significant European power via imperialism. Mixed political handling due to desire over mainland territory, and suffering from a reduced reputation due to poor war performance, led to no strong Italian claim on the diplomatic stage which meant that Tunisia went to the much more commercially invested France.

The reasons of expressed interest in Tunisia by the Italian government cited ethnic Italians who immigrated to Tunisia in the decades prior, with Italian communities mainly from Liguria and other parts of Northern Italy settling along the Tunisian coastline in the early 19th Century. These Italian communities were small, numbering less than 10,000, but served as a basis for a nationalist claim to Tunisia. They also cited historical Italian occupation in Tunisia, where the Republic of Genoa held a fortress in Tabarka along the Tunisian coast from the 1540's to the 1740's.

The more alluring reason for control of Tunisia however was control of trans-Mediterranean transit. Should Italy's Regia Marina have ports in Tunis as well as Sicily, any transit between the recently constructed Suez Canal and the strait of Gibraltar would have to pass through Italian-controlled waters, putting Italy in a position of strength in the Mediterranean that could act as a counterbalance to Britain's control of both entryways in and out of the Mediterranean.

Italy claimed Tunisia shortly after unification, and was supported by Austria-Hungary and Germany in order to offset French interests in the region, but France was quicker to act and made Tunisia a French protectorate, infuriating Italy and serving as the catalyst for Italy's entry into the Triple Alliance.

In a previous question I answered here, I touched upon France compensating Italy's loss of Tunisia with territorial promises of Libya in return which was recognized in 1902.

Control of Libya gave Italy good leverage over the trans-Mediterranean transit that they sought, but not complete control as Britain's control of Malta still meant that Britain had almost unhindered access across the entire Mediterranean.