A really good question and I'm often struggling with the same thing. Great answers too.
LFOs.. Sometimes you wanna go and add loads of automation on stuff, but sometimes it can be cool enough to have stuff modulated very slowly. Renoise has great freedom in this, but I've been working with Live for a while now which works slightly different. Still, "automated variation" can help: a filter cutoff or volume/pan modulated on dotted time intervals (3/16, 3/8, 3/4), or on frequencies not related to the beat (spin function can be fun, instead of phase).
Gear.. The variations killer is sometimes a workflow with mouse and keyboard. There are too many parameters that can be automated, and you have all the time in the world, seemingly, to get it "right". But all flow of direct creativity can be leaking away. Listen again, click click click, listen again.. Maschine is in my eyes a good step in the right direction for leaving you with *physical* rotaries for the important settings. I don't have it though, so what do I know. I'm just saying, getting physical gear really actually helps creativity.
Phase.. Jumping to a new phase with your song. I've watched an inspiring youtube vid where somebody said - traditionally there are 5 phases to making a song (Sound design, Recording, Arrangement, Mixing, Mastering) and even though you can try to do these all in one session it helps to have these 'barriers'. So to make a boring beat interesting you can try this. Render the tracks and load them back up in a new song file (call it "sick beat x - arrangement and mixing phase"). Line up an acapella (ideally that you know very well) and make sure it goes with the beat, the kick and the snare, exactly like the original. Bind various midi keys and knobs to simple things in your song - track volumes, mutes/solos, filter cutoffs. Delay times and amounts if you feel like going dub. Do one round recording of those parameters. Glitch combo fx plugins can be great for this too.
NoQuantize.. Vary by playing/programming in drumbreaks instead of loops. If you have *any* sound layed down yet, turn off the metronome too. At least for me, I'd rather record drums in against a steady 8th note hihat than against that *beeep* beep beep beep. Sometimes I get the illest breaks by fingerdrumming into a multitrack loop recorder.
Beatboxing.. Beatboxing is my favorite tool to come up with new grooves or breaks. Sometimes a beat gets way cooler when you just switch one bars drum pattern up to kick-snare-snare-snare. Like a sort of 'error' retrigger. So whenever you're on the bus or sth doing nothing but listening your fav beats, beatbox along with that (I suggest not to loudly, or perhaps just in your head). Then when you're home try to program that exact beat. Then load up the song and align the hits to perfect it, so you see how much 'shuffle' goes into a groovy beat. It's also fun to just record your beatboxing and turn that into good sounding drums.