一眼之念 is an imitation of the structure
一A之B. The structure usually works in the way of
noun(B) of a noun(A):
一面之缘 Fate of an encounter
一饭之恩 Kindness of a meal-offering
一国之君 King of a nation
Personally I don't think
一眼之念 is a particularly good imitation. Because it would be interpreted as
Longing of one look, and like the English expression I've used, it is similarly unidiomatic and ambiguous. However, it's pointless to blame a pop song for being ungrammatical, and it's easily understood if we put it into context.
一念执着 is a shortened way of saying
一种执着的念想. In English it would be
An obsessed longing or we could use a coma and write
A longing, obsessed. In Chinese it could also be
执着一念. You asked about grammar, but in fact you can do all sorts of word play and ignore most of the grammatical restrictions in Chinese poems.
So to sum up:
一眼之念 A longing, result of one look
一念执着 A longing, obsessed
Not as beautiful as @NS.X.'s interpretation, but it might be closer to the original meaning.