There is a tenuous connection, but it is not nonexistent unlike Ghana (named in honor of the empire of Wagadu by Nkrumah's CPP) and Benin (changed from Dahomey for historical and political reasons in 1975 to a geographical name based on the bight, not specifically to appropriate the name of the still-extant kingdom of Benin in Nigeria). The heartland of Mande, and particularly Mali, is in the far southwest of the country, where Bamako and other towns are; much of Mande is however also in Burkina Faso (former Upper Volta) and Guinea. So there's not a linear connection, any more than there's a linear connection between it, modern Mali, and Samori Touré's Mandinka (Wassoulou) empire that the French toppled. A lot of states emerged and covered the same areas in the roughly 400 years between the shrinking of Mali's effective power to the Niani region and the arrival of the French, so any direct restoration claim would have been very difficult.

There is, however, a lot of implied cultural connection, and a claim to the name turned in part on the incorporation of historic holdings of the empire. Initially, Mali was the "Soudan Republic" (from French Soudan) and formed a "Federation of Mali" with Sénégal (also in its former holdings) in 1959 [edit: 1958]--possibly as a direct call to the past in the mould of Nkrumah's naming of Ghana. That federation did not last much past independence owing to the personalities involved in Bamako and Dakar, but it's worth pointing out that the name also involved drawing in further parts of the former empire. The name, however, stayed with the modern Republic of Mali.

It's worth noting that the first President of Mali (well, the Soudan Republic) was a Keïta, as is the current one--the same family name as the ruling house of the mansas of Mali--much as the first head of state in Guinea (1958) was a descendant of the ruler of the Mandinka Empire, Sekou Touré. This is further cultural continuity in the region, and the names that appear in Malian politics tend to replicate the great families of Mande as well as the northern bend of the Niger--the Haïdaras (of whom one branch has been instrumental in manuscript recovery in Timbuktu), the Traorés (after the military coup of 1968), and others. So there's no linear descent, but much of the same area is covered and there are definitely overlaps that the state benefits from honoring, because there are (or were, before Ansar Dine) tourism and international-heritage funds to be had. Besides, there's a good stratum of educated Malians today who appreciate the depth of the knowledge they have in their possession, and that's a valuable thing for further buttressing a sense of national heritage.

To answer the question simply, then, it was the creation of a new nation--one heavily dominated by Westernized elites--but one that also has a use for the older cultural link. There are no mansas, and no serious movement for cultural Africanization (as in, say, Mobutu's Zaïre), but from my experience the sense is still important. I'm not sure how actively the connection has ever been exploited simply because not everyone in the state of Mali today exactly had a reason to celebrate that empire, but the value is not zero.

I will echo /u/Khosikulu's characterization. There is a tenuous connection there, and the modern Republic of Mali can point to cultural sites like the Sankore Mosque and Djinguerber Mosque in Timbuktu as a cultural linkages. It can also be mentioned that modern Mali has many speakers of Mande languages (Mandinka, Dyula, Bambara), whose ancestors had prominent places in the Mali Empire.

However, there are some strong reasons not to view the modern Republic of Mali as a "rebirth" of the Manden Kurufaba. From the time the Songhai rose up to challenge Malian supremacy in 1430 until 1960, there have been many political entities that have exercised control over the middle niger region. There is the already mentioned Songhai empire, but also Bambara states at Macina and Segou in the 18th and 19th century, and the Fula led Tocoleur Empire of the mid-19th century. Of course, the French exercised control for 90 years until 1960.

This period of French rule did quite a lot to transform the region and the people, and it had a lasting impact on post-independence Mali. French continues to be recognized as an official language. Administrative régions and cercles originate out of French colonial practice. Mali is a republic, with organs like a parliament and a president influenced by French ideas of Republicanism. /u/Khosikulu has rightly pointed out that several powerful figures in recent Malian history come from families/lineages that were prominent in the Mali Empire, but there has not been any move to revive the institution of the mansas, or in other ways "revive" the empire in a meaningful political way.

I would agree with his assessment that the name Mali and the consequent associations with the historical Mali Empire has been useful as part of a national myth, as a reminder that the region was wealthy, sophisticated and powerful once, and as a promise of future wealth/glory/power/sophistication.

Very interesting. I'd imagined that the connection would be like that, loose but still existent.