Alright, let's untangle this bit of myth-mash: the short reality is that the old Norse sources never flat-out call Loki a 'frost giant' the way modern pop culture often does.
In the poems and prose that make up the medieval corpus—especially the poems collected in the 'Poetic Edda' and the handbook-style 'Prose Edda' by Snorri Sturluson—Loki is presented as a complicated, liminal figure. His father is named Fárbauti, who is explicitly a jötunn (a type of giant), and his mother is Laufey (sometimes called Nál). So yes, Loki is of giant blood: he’s the son of a jötunn, but he lives with and is counted among the Æsir (the pantheon of gods), causing all the delicious ambiguity. Texts like 'Lokasenna' and the genealogies in 'Gylfaginning' show him moving freely between roles—guest, trickster, friend, enemy.
Where the idea of a 'frost giant Loki' really comes from is modern adaptation, most famously the Marvel universe, which reimagines Loki as literally born of the frost-giant race of Jotunheim and adopted by Odin. That version leans on the jötunn heritage but flips parentage and emphasis: in Marvel, Laufey is male and a frost-giant king. The medieval poems don’t make that gender swap or label Loki as a member of the hrímþursar (frost-giants) specifically. So if you want the canonical Old Norse take, think: giant parentage, but socially and narratively mixed into the gods, not a straightforward 'frost giant' identity. I love how the ambiguity fuels so many retellings—Loki is perfect for reinvention, and both the ancient texts and modern comics get mileage from that edge.