Hybridisation of oxygen in diethylether is

In diethyl ether oxygen undergoes sp³ hybridisation forming four sp³ hybrid orbitals.

Diethyl ether's molecular structure is ${\mathrm{C}}_2{\mathrm{H}}_5-\mathrm{O}-{\mathrm{C}}_2{\mathrm{H}}_5$.
Here, the oxygen atom in the center is ${\mathrm{sp}}^3$ has two lone pairs of electrons hybridized.

Hence, the correct option is option (C) ${\mathrm{sp}}^3$.