In recent years whenever I or my wife gets a prescription, the vast majorty of them seem to list among their possible side effects, “dizziness.”
I would not be surprised if medicine to help with dizziness also lists that as a side effect. 
Why do so many medications cause this side effect?
What physiological mechanism is involved?
If it’s medication to lower blood pressure, the low blood pressure itself might cause dizzyness.
'bout all I’ve got.
There are a lot of different ways to get to “dizzy.” Dehydration, lowered blood pressure, vestibular effects, vision changes, effects on the cerebellum, etc.
Shoshana is very astute for placing quotation marks around “dizzy” and (I believe) is making the point that “dizziness” means different things to different people. Therein lies why it’s such a common side effect, i.e. you’re not talking about one, particular side effect but, rather, a number of different ones all lumped under the name “dizzy” or “dizziness”.
There are at least four things that people can mean when they say they are “dizzy”:
Light-headedness with standing (orthostatic hypotension) can be a side effect of any medicine that lowers blood pressure and/or that causes salt loss (e.g. diuretics). Orthostatic hypotension also occurs with medicines such as nitroglycerin and other ‘nitrates’, certain anti-depressants, various tranquilizers, anti-Parkinson’s Disease medications, and many, many others.
Generalized weakness, whether subjective or obective, can probably occur as a side effect of any drug. Some of the more common offenders in this regard are medications for high blood pressure, angina, antibiotics, and drugs used to treat “auto-immune” diseases and cancer. As noted, though, taking almost any drug can make you feel weak.
Unsteadiness, or clumsiness, can complicate the use of any medicine that affects the central nervous system. This would include essentially all tranquilizers (“major” or “minor”), antidepressants, and anticonvulsants. Narcotics would also fall into this category as would many other drugs.
True vertigo is fairly rare as a drug side effect. Taking too much Lasix or aspirin might cause it. Likewise, vertigo can develop after taking certain antibiotics (e.g. gentamicin) for too long and/or at too high a dose. Of course, alcohol can cause vertigo as well.