Another option that I think I learned at a L and L conference is to completely ignore the child. First tell them that you can't hear a whiney voice and then completely ignore what they're saying. I found this too difficult to do and so I send/take them to their room when it continues and I'm unable to stop what I'm doing. A note here. Sometimes when I'm also tired and feeling cranky I choose to not stop what I'm doing..
An insight that I gained with my grandchildren is that I whine when I'm tired and/or hungry. Since then I ask if they're hungry before I try to deal with the whine. They usually say no, accept a snack, or get worse because I won't give them a piece of candy. :):)
So my method is to take them to their room, put on some quieting music or even turn on the TV, putting in a quiet CD, hand them what ever is their current stuffed animal with which they sleep and leave. I see this as a literal time out from life and not as a punishment.
Sometimes I check back and they've been sound asleep. If it's late in the afternoon I wake them up so that they can sleep at night. Or I might give them dinner and put them back to bed. When that happens they usually sleep thru the night.
My thinking about turning on the TV is that watching TV is a way to escape and relax.(zone out). It reduces active stimulation. If they're not overly; tired they usually start playing or just come back out once they've rested enough. If they're still whining and I'm able I pick them up and try the rocking or cuddling again. If that still doesn't work I take them back to their room.
When I'm able, I just pick them up and hold them for awhile. Then I try out another way if that doesn't work. When they're overly tired they often don't want to be held.
I also noticed that when I've been whining or am just tired and not paying attention to them and they start whining that holding them and focusing on relating to them works best. It also gives me a break. I think kids pick up on a negative change in our mood and not knowing what else to do start misbehaving to test the situation. Now that my granddaughter is older (she's 9) she is checking in with me more by asking if I'm mad at her. What a relief that is for me.