/u/sagathain and I recently replied a similar topic in: Why are most sources of Norse mythology from Iceland?

To complement a bit on their literary activity in general (not confined to famous mythological stuffs), it is true that the medieval Icelanders were generally prolific, that is to sat, those who recorded more in parchments, at least compared with their contemporary Scandinavian fellows, and this phenomena can mainly explained by two different ways:

1. Willing acceptance of vernacular (Old Norse) literacy by local Icelandic elites, as I ilulstrated in the linked thread: In spite of their later acceptance of Christianity and Latin alphabet culture attached to it, Icelandic elites could largely catch up the contemporary European trend of the flourishment of vernacular literature sponsored by aristocratic families like chivalry literature. In short, high and medieval vernacular literature and its production often served to cultivate the elites' cultural identity, as argued by Spiegel (Spiegel 1990), and the prolific production of Icelandic sagas largely also fell into this pattern. Icelandic elites also knew and made use of such a possible value of literature.
2. In course of changing social circumstances and their relationship with the outer world and authority like the king of Norway mainly from the 12th to the 14th century, namely the more vehement power struggle among the Icelandic elites (sometimes known as the Age of the Sturlungs), the subjugation to the king of Norway, and their gradual isolation from the changing nominal sovereign (the king of Denmark had assumed the rulership to Iceland since the establishment of Kalmar Union (1397-1523)), medieval Icelanders (especially their elites) kept on producing the different sub-genres of sagas to assert their own cultural identity against these political and social changes (Jón Viðar Sigurðsson 2010). To give an example, the composition and selection of more than 200 literary works collected in the largest medieval manuscript, Flateyjarbók, is now analyzed from the point of view that how the Icelanders saw their past as well as their changing surroundings (Cf. Rowe 2005).

Or is it just that their production gets more attention?

On the other hand, you also have a good point to pay attention to post-medieval reception and popularity of medieval Icelandic literature.

Since the 17th century, two early modern Scandinavian powers, namely Denmark (whose king was also the king of Iceland) and Sweden had competed each other to hire the Icelanders and to salvage 'their' cultural memory from the dusted medieval Icelandic parchments. From this competition to secure 'Scandinavian cultural capital', the manuscript of Eddas was found and dedicated to the king of Denmark by the Icelandic bishop, and the Danish (Dano-Norwegian) and Swedish translations of the most famous kings' saga (royal biography of ancient and contemporary Scandinavian rulers), Heimskringla, had also already been translated and published in the 17th century. Thus, despite of a few large-scale city fire in early modern Copenhagen and destructions of the manuscript collection, medieval Icelandic manuscripts had attracted much more attention from the antiquarians, and sometimes been treated better for long, at least than some of their European counterparts.

After Iceland became independent again in the middle of the 20th century, after more than 650 years after their subjugation to the king of Norway in 1262/3, where to store medieval Icelandic manuscripts, 'their' (who?) cultural treasure, soon became the cultural as well as diplomatic issue between Denmark and Iceland. In 1971, the codex Regius of Edda was returned from Denmark to Iceland after more than 3 centuries' absence from its homeland.

(added): The following link of University of Copenhagen offers a bit details on the 'transfer of medieval manuscripts to Iceland' (from a Danish point of view) in the late 20th century: https://nors.ku.dk/english/research/collections/arnamagnaean_collection/transfer_iceland/

Add. References:

• Gísli Sigurðsson & Vésteinn Ólason (eds.). The Manuscripts of Iceland. Reykjavík: Árni Magnússon Institute in Iceland, 2004.
• Jón Viðar Sigurðsson. 'Historical Writing and the Political Situation in Iceland 1100-1400'. In: Negotiating Pasts in the Nordic Countries: Interdisciplinary Studies in History and Memory, ed. Anne Eriksen and Jón Viðar Sigurðsson, pp. 59-78. Lund: Nordic Academic Pr., 2010.
• Rowe, Elizabeth Ashman. The Development of Flateyjarbók: Iceland and the Norwegian Dynastic Crisis of 1389. Odense: The University Press of Southern Denmark, 2005.
• Spiegel, Gabrielle M. 'History, Historicism, and the Social Logic of the Text in the Middle Ages'. Speculum, 65, no. 1 (1990): 59-86. Accessed August 15, 2021. doi:10.2307/2864472.