I was thinking about this question. And, tonight I had a few hands where I raised a flop c-bet (for value). First hand, we had deep stacks:
10nl -- 4 Handed
BTN: 41.5 bb
SB [Hero]: 170.6 bb
BB: 30 bb
UTG: 205.2 bb
Pre-flop: (4 Players) [1.5xbb] Hero is dealt
UTG raises to 3xbb
, 1 fold, Hero calls 2.5xbb
, 1 fold.
Flop: (2 Players) [7xbb]
Hero checks, UTG bets 3.5xbb, Hero raises to 15.5xbb
, UTG calls 12xbb
I need to raise here because we're 170xbb deep, and I won't be able to play this for stacks without getting 4 bets in after the flop. I want my last bet to be about 2/3rds of the starting stack (at most). This is because that will be a pot sized bet. That means I want to bet about 114xbb on the river.
He put in 6.5xbb, that's a lot of money to go (49.5xbb by the river). So I divide the 50xbb by 4 and raise that amount here. I will bet the rest of the 50xbb on the turn.
This assumes he calls. But, on this flop, I am not afraid to raise here. If he folds, I probably wasn't going to make much anyway (hands that fold to a raise here are just too weak to play for stacks). If I raise, I get massive value out of trips. I really want him to have A-8 here, because he'll never fold that.
Turn: (2 Players) [38xbb]
Hero bets 36xbb
, UTG calls 36xbb
Turn card changes nothing. I continue with my plan and try to get the rest of the 49.5xbb in (I bet 36 -- which is a hair low). He calls, which is great because I am set up for my river shove.
River: (2 Players) [110xbb]
Hero ALL-IN for 116.1xbb
, UTG calls 116.1xbb
This is not the best river in the world. But, there are a lot of 8-x hands other than 9-8 in this range. If he's sucked out on me, he's going to win a big pot. I have just a little more than a pot sized bet (6.1xbb -- 61c here), and I stick it all in.
UTG thinks about it for about 20 seconds, and makes the call.
--- Full House, 4s full of 8s.
--- Three of a kind, 8s.
Hero wins: 332.2xbb (10xbb rake).
So, in this situation, I am raising for value. I need to make my raise large simply because of the stack sizes. If I don't raise large, I can't get stacks in by the river.
With smaller stacks, say he had 50xbb at the start of the hand, I have more options. I could raise to (13.5xbb) and shove the turn (33.5 into 34xbb). Or I could min-raise to 7xbb, bet 14xbb (2/3rds pot) on the turn, and bet 26xbb (about half pot) on the river. Those smaller bets might suck in some more suspicious players.
Same hand, same goal (getting the stacks in), but different bet sizes based on his stack. So, the answer to your original question is very complicated.
This is just one hand. If you have a weaker hand, where you might not want to get stacks in, you would make different bets. If you have a strong draw (combo-draw), you might raise more to increase your fold-equity. If it's a pure bluff, you want to raise enough to get the folds when he's c-betting air but no more.
And that's just a few considerations. To be honest, my biggest concern, if I am raising a flop c-bet, is usually setting up shoves. I may not actually follow through with that plan, depending on the turn and my actual hand strength. Still, a big way to increase your wins in NL is to get someone's stack more often. To do that, you need to be setting up your bets in a way to win stacks.
Edit: I had two other hands (one a bluff and one where I was raising a donk-bet that I figured was weak but I would have to fold to a reraise) but I lost both of them because of Bovada's horrible hand history interface. I might be able to find them tomorrow or the next day, when I import them into PT4. For right now, I only have this one.