Could LEP II have discovered a 125GeV Higgs? LEP II eliminated the Higgs up to 114.5GeV. If it had been run for longer could it have detected a Higgs at 125GeV?
I Googled for this without any luck, though I did find a comment that LEP II topped out at 209GeV collision energy, so it seems as though production of a 125GeV Higgs would have been possible. If so, how much longer would it have had to run?
 A: As @Guy mentioned in their answer, the main limits were set looking at the production method $\def\fs#1{\rm e^+e^-\rightarrow #1}\fs{ZH}$. This production method has a large cross section once an energy threshold is met. For the Standard Model Higgs as discovered at the LHC, this threshold is about $\sqrt{s}=217~\rm GeV$. So close!
Other production methods have a much smaller cross section once this threshold is passed. But unlike $\fs{ZH}$, they can still occur below the threshold. Since their cross section is low, LEP II would have to run for quite a long time to detect the Higgs boson. But it could have been done.
Resonant production
A generic process $\rm x\bar x\rightarrow y$ receives a large enhancement in its cross-section if the center of mass energy of $\rm x\bar x$ is very close to the mass of $\rm y$. For the interaction $\fs{H}$, this brings the production cross section all the way up to around $.5~\rm fb$. At that rate, LEP II could have produced around one Higgs boson every two years!
Of course, there are problems with this. First and foremost, running like this would have required the experimenters to know the Higgs mass a priori. Actually running the machine at the precision required to take advantage of the Higgs resonance would be difficult. And even with all that, it would still take a long time to actually discover the Higgs boson. So this would not have been a realistic method of discovery.
Gauge boson fusion
There is one more significant Higgs production method at LEP, called "gauge boson fusion." This is a process in which the electron and positron each emit either a Z boson or a W boson, and the two merge to form a Higgs boson. So, one of the following processes:
$$\def\mfs#1#2{\fs{#1}\rightarrow#2}\mfs{\nu_e\bar\nu_eW^+W^-}{\nu_e\bar\nu_eH}$$
$$\mfs{e^+e^-ZZ}{e^+e^-H}$$
According to this paper, the former of these two processes actually has a significant cross section at $\sqrt{s}=209~\rm GeV$, of about $2~\rm fb$. LEP II actually delivered enough integrated luminosity that it's fairly likely that it actually did produce a single Higgs boson some time during its run. If the LEP II had run long enough, this would have eventually been the discovery channel. 
"Long enough" is a long time, though. Given that this cross section is about 100 times less than $\fs{ZH}$ becomes past the threshold energy, and that $\fs{ZH}$ probably would have sufficed for discovery within a year or two, a quick estimate is that it would have taken 100 years to discover the Higgs this way. (This estimate is rather optimistic, since although running longer increases the signal strength linearly, it also increases the background, so it would likely take much longer). So the LHC was definitely a better way to go.
A: The LEP experiment's limits on the Higgs mass were set by looking for a process where the experiment would have produced a Higgs boson together with a Z boson.  The highest energy they achieved for the electron-positron pair which annihilated to make Z,Higgs was 209 GeV, and that was only achieved in the last months of the experiment.  Since the Z boson mass is 91 GeV, the highest energy Higgs boson which could be produced this way would have a mass of 209-91=118 GeV.  Some of the energy is always lost to getting the Z and Higgs to move apart from each other, so in practice the limit they could achieve was a little lower than this, 114 GeV.  By running much longer and accumulating statistics they could have extended their reach a little bit, perhaps to 116 GeV; but not to 124 GeV.  That could only have been achieved by significantly increasing the energy of the beams -- which I believe they had already pushed as far as they could.
