Meld into stone is a touch-range spell that melds the caster with the stone that he or she touches for the duration of the spell. When the spell ends, the caster exits the stone at the place entered. (See Can the Meld Into Stone spell be cast on someone else? for related.)
One way to deliver touch-range spells is through the touch of one's familiar.
What would happen when a caster delivers a meld with stone spell through the touch of their familiar?
I can imagine three possibilities, none of which seem very appealing:
The familiar touches and melds with the stone. Nothing happens to the caster.
The familiar touches the stone. You (the caster) meld with the stone at the place that the familiar touched, assuming other requirements are met (notably, 100 foot maximum distance between you and the familiar.) At the time that the spell ends, you return to the exact place that you were when you cast the spell, rather than adjacent to the wall you entered.
The familiar touches the stone. You meld with the stone at the point of the familiar's touch. When the spell ends, you exit the stone at the location that the familiar occupied at the time that the spell was cast.
Of course, there may be some other possibilities I've missed.
A few other situations seem related. One is casting plane shift through the touch of a familiar. Another is casting meld with stone with the distant spell metamagic.