I recently read "La Chanson de Roland" and "Le Couronnement de Louis/Charroi de Nîmes" in my medieval french lit. class. In both of those chansons de geste, both dating from the ~10th and ~11th century there are many named swords that are featured in the works.

For instance, there are a number of passages about Roland's sword, Durendal. Also Charlemagne's sword Joyeuse, and the villian in "Roland" has a sword names Précieux.

Because I'm more familiar with the medieval literature I cannot speak specifically to actual real life documented examples of people naming swords.

However, due to it's inclusion in the aforementioned chansons de geste (and in other works of that genre) it was likely an important aspect of medieval society. One can make this conclusion because of the fact that throughout works of that genre you saw the authors emulating their society, or at least an ideal of there society. Of which naming swords were important.

Also, just to add on. Perhaps someone here can give an estimate in modern money, but swords were extremely expensive possessions in the Middle Ages. They were extremely important to the owner because of the sheer cost of owning one. This perhaps influenced why names were given to swords.

Warriors not only named their swords, but they also had them blessed in formal religious rituals, and inscribed them with prayers (“Jesus and Mary,” “Hail, Mary full of grace,” “God is the conqueror of all. Amen”). One late 13th-century has the inscription “Amor vincit omnia” on it: Love conquers all.
Source: all of these examples are from William Chester Jordan’s 2009 lecture “Crusader Prologues: Preparing for War in the Gothic Age.”

I recently read most of the famous Norse Mythology stories, and I was surprised by the fact that almost all tools/weapons/houses/horses had names. Your question was about people (and these were gods), but people wrote these stories, so the idea of naming swords had to exist.