(for pure speculation and reflection)
I think it's most likely Tirhut.area in northern Bihar, neighboring eastern Nepal.
Thanks Dan, but actually I am interested more in it as a lexeme and if the name is attested in any Sanskrit source. So what would be the Sanskrit name?
OK, D., There are epigraphical sources for the name Tirabhukti, of which Tirhut is a much-reduced form, resulting from at least a month of fasting. The works of Sircar are the place to look. I'd look there, but I have to get biking. Have a great day! —D
Thanks Dan! OK, at least the name “Tīrabhukti” is recorded in MW (s.v.) and explained as “m. Tirhut (province in the east of central Hindustan) L.”