Well, having just watched the series, I think they have the right of it, in that Tohru hopes that by having Miss Kobayashi eat her tail, she will gain her power, thus allowing her to keep their little family together.
Not only is there the concept in parts of Asia and most early tribal civilizations of eating mythical beasts to gain their power... but then there it the Ouroboros myth, the dragon eating it's own tail representing eternity.
I think it's also very likely, given Tohru's feelings for Miss Kobayashi, and also knowing that her sexual advances have been rebuffed so far, that it also would represent a kind of physical bond between them, a way for a little part of her to be inside Miss Kobayashi, and therefore have some bit of intimacy through that.
In summary, I believe the reason is twofold: a combination of a desire for a physical bond between them and a way to share her immortal lifespan with her so try to be together, if not forever, then for a long, long time to come.
It's a bit odd from a human perspective, but in Tohru's mind it would be a bit romantic. Like a mate that kept a lock of your hair in a locket around their neck all the time so they'd always have a piece of you with them.