I'm DMing a game and one of my players is a Warlock of the Great Old One. I want to send him some cryptic dreams and maybe some demands from his patron, but I don't want to get too dark (like I don't want him to be required to kill anyone etc.) as it wouldn't really fit with the tone of our game. Does anyone have any suggestions?

For GOO, I could see the patron sending visions compelling the warlock to perform some odd ritual, or build something.   If you've ever seen Close Encounters of the Third Kind, imagine Richard Dreyfus building a replica of Devil's Tower in the living room.

Where the pacts of the Fey and Fiend would likely ask specific tasks that the warlock knows the result, it would be fun if the requests of a GOO seemed just strange at first and not really apparent what the request actually accomplishes.

Quote from socon67 >>

Where the pacts of the Fey and Fiend would likely ask specific tasks that the warlock knows the result, it would be fun if the requests of a GOO seemed just strange at first and not really apparent what the request actually accomplishes.

 Basically this. Make the Warlock perform tasks that appear irrational, nonsensical or simply useless.

Have them buy an expensive doll only to throw it away a few streets away. Make them switch out items without other people noticing. Sometimes it's enough just to play messenger to a person you've never seen, delivering a message that doesn't make sense to you.
The perspective of a GOO patron cannot be compared to that of a simple warlock. For them, the warlock is just a simple pawn; a servant to do the smallest tasks they don't bother with.

EDIT: I forgot to mention: as soon as you make the player realize what consequences their seemingly simple actions have, their brain will explode. It's enough if they just realize the consequences of a single task, to make them more interested in what the other tasks could have for consequences.

So when I rolled on the random flaws for my character, I forget the exact wording but essentially he is unreasonably horny. He seduces everything and anything anytime they are in a town. My DM  decided to play with that as part of my Warlock Pact so the Archfey I am sworn to basically drains/absorbs all pleasure and memory from my PC while he is ahem getting it on. The DM has used this for some interesting moments when I meet those same NPC’s at other times and I don’t acknowledge or know them.

I have a warlock I my game too, so this has crossed my mind.  For mine, I thought maybe have his Patreon tell him to simply convert people, but if you want something more goolock-y, maybe have him be asked to eat raw calamari or maybe tell him to maybe wear really odd clothing

Make the player adopt a suitably mysterious cat. 

Here it is 2022; I would miss the post when it was originally posted. My best advice for anyone running a warlock with a GOO patron is to read up on all the Call of Cthulhu material on the Internet. The GOO concept was based on Lovecraft's Cthulhu mythos and there is a massive amount of info available on the web. I even found the one "good" GOO for my Neutral Good Paladin/Warlock multiclass. Well, "Good" in a GOO kind of way. "Creepy Good" is more like it, which fit my NG Paladin of Vengeance/Warlock rather well.