What exactly does "for the love of God" mean?

Solution 1:

"For the love of God" is definitely, as Laetitia was getting at, an expression of desperation. If you exhort someone to do a particular thing, and use this phrase, you mean

Don't do it for me. Don't do it to make money. Don't do it to protect your business or your good name. Do it for a more basic motivation, the motivation that we were all born with, namely, the love of God.

It means that the thing you're exhorting the person to do doesn't need fancy motivators, and shouldn't need much convincing. It means you are at the end of your rope (i.e. you have run out of patience).

Suppose someone has been going back and forth for a ridiculous amount of time, arguing with himself about whether to do a particular thing or not. If this goes on indefinitely, at some point you are going to blow up and tell them to just go ahead and do it, "for the love of God".

For example, "Should I ask her to the dance?" After half an hour of prevaricating, you dial her number, thrust the phone at your friend, and shout, "For the love of God, just ask her already!"

Solution 2:

It's an expression of frustration and anger.

http://idioms.thefreedictionary.com/for+the+love+of+God

An oath of shock, exasperation, annoyance, frustration, or anger.

• For the love of God, I didn't even see that car coming!

• Would you let me finish my story, for the love of God?

• Oh for the love of God, I just had the car fixed and now you've put a dent in it!

Here is an explanation of the the for the love of part:

(http://idioms.thefreedictionary.com/for+the+love+of)

for the love of : For the sake of; in consideration for: did it all for the love of praise

Hence, it seems that originally it was closest to your second suggestion.