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The problem is caused in the first place by a mistake in NumberFieldFactory. In line 610 of sage/rings/number_field/ 6.9), the latex_namedoesn't get passed on to the extension method of the base field.

Unfortunately, you cannot get round this by using extension directly; in this case:

sage: G.<alpha> = NumberField(x^2 - 2, latex_name='alpharedefined')
sage: print latex(alpha), G.latex_variable_name()
alpharedefined alpharedefined
sage: P.<t> = G[]
sage: H.<beta> = G.extension(t^4 - alpha, latex_name='betaredefined')
sage: print latex(beta), print H1.latex_variable_name()
\beta betaredefined

The inconsistencies here indicate that there are other things that need correcting in this part of the code.