The usual choices aren't made over the translation "burnt offering." Mosaic law followers had practices not required by later Jews or in the NT (religion as law of the land with death penalty, even cherem, etc.) but the story of Jephthah has issues with other parts of Mosaic law regardless notably Mosaic law banning followers from sacrificing their childen (Ps.106:35-38; Is.57:5).
See the articles at the next links for a fuller explanation than I'll give here. http://www.jba.gr/Articles/jbadec06b.htm http://en.wikipedia.org/wiki/Jephthah#Sacrifice_controversy
The usual translation alternative is due to the fact that "...the connective particle ו (vau, our English v) is often used as a disjunctive, and means 'or', when there is a second proposition."
This creates an "alternative.... He would either dedicate it" (whatever came out of his door) "to Jehovah (according to Lev. 27), or, if unsuitable for this, he would offer it as a burnt offering."
Jephthah is given as having strong faith (Hebrews 11). The choice is that he was either out of accord with the usual understanding of Mosaic law banning followers from sacrificing their children--a personal rash exception with tragic consequences--or that he dedicated his daughter to virginal service to God. There would be something for the daughters of Israel to lament either way or rehearse with her if she lived (Judges 11:39,40).
