Results will dictate that. There's no point blaming the fans either. At one point or other during the last six years, each and every fan probably wanted to give a particular manager time while the majority of the fanbase bayed for blood (whether that was Roberto, Koeman, Silva, Big Sam, Benitez, Carlo, or Frank). In the end, the mob got its way (rightly or wrongly). Many fans will, therefore, expect the same standards to apply to any manager who they are not emotionally invested in.
My position on Frank is clear. I don't particularly rate him but know he is not the problem. As an Everton fan, I wish him only the best. But if he runs into a run of untenable results a la Benitez, well, I won't be manning a stout defence of the man's continued employment. I know the problem is in the board room. Lampard is almost beside the point. Any manager is. This is not a squad that can be expected to do anything more than scrap for survival. So, in that sense, Frank really just needs to avoid regular thrashings at the hands of the bottom half of the table and keep us relatively comfortable in lower mid-table to hang on to the poison chalice, I mean, job. If he can't do that, well, who cares really.