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I wouldn't attempt it for several reasons:
1. the signals feeding an LCD are very different to those used with an LCD. For example you will not have individual segment conections and digit drive signals.
2. it looks like you have one (1.5V) or two AA cells (3V) which isn't enough to drive most LED displays. Each segment in a standard display has several LEDs in series which means they need higher voltages to turn them on.
3. 7-segment LEDs are physically much thicker then an LCD so you would have difficulty fitting it in the same box.
If you really want an difficult but interesting project, you could build an entirely new LCD clock into the box but to be honest, it would cost several times more than buying a new one in a store.
Brian.