Eliminating the Lt. Governorship in general seems like a pretty sensible platform position. There are exceptions where the Lt. Governor wields real power, such as in Texas, but plenty of states have no use for them.
My idea is as follows:
Lieutenant Governor should be abolished as an office, but kept as a title that would be held in conjunction with other elective office. The Lieutenant Governor would be appointed by the Governor from among elected cabinet members or state legislators, and would be confirmed by the legislature.