Paying a contractor to be "on call"?

We're looking at hiring a systems administrator contractor at a certain hourly rate to do sysadmin-y things. So far, this is easy enough for me to wrap my brain around.

But during certain hours, we'd like this individual to be "on call" or on standby. He won't be "working", but the expectation is that he will never be more than 15 minutes away from an SSH prompt during those times.

Should he need to spring into action, his normal hourly rate applies of course, but what about when he is on call?

Is it customary to pay a contractor a reduced rate to be on standby, or do hourly sysadmin contractors normally expect a certain amount of 'on call' time as part of the job?

The company and contractor are both located in the US.

Well... in US you can't dictate to 1099 when you want them to be on-call, only to a W2. States love to do audits of little companies and you will not survive this one. Workaround could be to offer a retainer for being on call. Use it or lose it kind of setup.
My comments are null and void, if you are contracting someone from outside of US, since I never dealt with that kind of setup.

It is based on your agreement with the person, although in some places governments mandate certain things as well. As you don't tell us what country are you at - it is very difficult to estimate what regulation might become relevant.

On a personal note, I would expect to be paid, if you want my time. It is your job to decide what I should do, and if you tell me to wait by the phone/keyboard - I'll wait, and would still be expecting to get paid for it. So personally, as such contractor, I wouldn't accept a difference in pay between my time being spent on you waiting for you to decide what to do, and my time being spent on you doing something.

How often do you have them be on-call?

I do the contract-IT thing right now, and I just bake in that ability as part of my hourly fee - since I'm expected to be somewhat available during the weekday, and since my 3G plan (that I don't use for anything else) will be rather expensive in a few months, I expect to be reimbursed for it, even if it's in a roundabout way, so I charge (gasp!) for it, albeit spread out. I'll swallow some of the cost, it's just the price of being available to do business.

So to answer your question: It depends. For the small network I help maintain, it's small enough that my phone beeps about twice a day - something goes down, and then it comes right back up. So I'll spend about 20 minutes a day outside of the work period in a bit of a "oh crap what-is-going-on" mode.