Interesting theory. Assuming swords also become weaker if they don’t have names aligns with my random theory that Higekiri isn’t as silly and scatterbrained as he seems.
According to the legends, the emperor cut a naginata into two equal halves. One half was given to Kokaji Munechika (Higekiri’s blacksmith) and another half to Moufusa (Hizamaru’s blacksmith). When the blades were presented to the emperor, Munechika was arrested because Higekiri was much shorter than Hizamaru; the emperor thought Munechika cheated him. Munechika prayed for a miracle and Higekiri got up from its stand “seemingly of its own will” and cut Hizamaru down to the same size, hence the name Tomokiri (“Friend Cutter”). He does this again to a replica of himself, which was given to appease him when Hizamaru was taken away from his side after many generations.
Higekiri claims names don’t matter much and that he’s forgetful because of the passage of time, but it’s sort of suspicious because he clearly remembers all of his names with how he alludes to them. In one of Higekiri’s citadel lines, he states “It's not good to be jealous of others. You'll become a demon...” which refers to a myth of him and Watanabe no Tsuna cutting a jealous demon down, but what if it also refers to him as well? Not to mention that he repeatedly assures the saniwa that he won’t cut people or things down (“No, no, I'm not going to cut down my allies just because I'm not the captain.”; “It’s okay, I’m not going to cut off their manes or anything.”). It seems he’s also aware of the power of names (“I wonder if a sword's name is something that contains the emotions of its owner?”).
So combining all of that Higekiri might be forgetting Hizamaru’s name on purpose. [spoiler:lit]Or maybe this is just the kind of tinfoil hatting you end up doing after reading multiple evil Higekiri Genjicest doujins.[/spoiler:lit]