Are there any attempts in the literature at addressing the origin of the Higgs field? And, which lines of research that find it inevitable to address this question?
In quantum field theory, the Higgs field is – much like the electromagnetic field, the W-boson field, the electron's Dirac field, and other fields – an elementary entity that can't be decomposed to anything simpler.
This claim of course assumes that the Higgs field isn't composite. The mass of the Higgs boson around 125 GeV, nearly discovered at the LHC, strongly indicates that the Higgs field isn't composite.
So one simply has a Higgs doublet of fields, $h_1(x,y,z,t)$ and $h_2(x,y,z,t)$, at each point of the spacetime. They're quantum fields (with hats) which means that the energy carried in the waves upon these fields is quantized (it is effectively composed of particles).
The now-disfavored models would construct the Higgs field out of more elementary fields. For example, in the technicolor models, one doesn't get objects that are "strictly identical" to the Higgs boson but one may produce similar particles out of more fundamental fields and particles such as the techniquark fields. The construction of the Higgs boson out of techniquarks is fully analogous to the construction of mesons (e.g. pions) out of quarks in Quantum Chromodynamics.
In string theory, all fields – electromagnetic field, Higgs fields, electron field, and others – may be derived from more fundamental building blocks, namely strings (and branes) and the string fields that create them (if one uses the string field theory description). In this picture, a Higgs boson is a vibrating microscopic string. The typical size of the string is so short, however, that one can't probe the internal structure by direct experiments.