This variation is uncommon, but it's certainly not unheard of. I don't know of a particular name for it, and rather than get bogged down in the endlessly-analyzed lines, let's just take a quick, objective look at what this move accomplishes, and what it gives up.
3.Bc4 aims, as you said, at the
f7 pawn. If allowed to remain on the diagonal, this Bishop could provide pressure against the Black kingside in the midgame.
- It develops the Bishop quickly, allowing White to castle kingside immediately if he so chooses.
- It may avoid trading away the Bishop for Black's QN or QB like we see in the Canal-Sokolsky Attack (
3.Bb5+) or other variations where White's KB lands on
- It grants White some additional influence over the light central squares weakened by Black's first two moves (
- By developing to
c4 so early, this move appears to invite Black to play
...e6, immediately limiting the Bishop's scope and possibly requiring a future relocation.
c4, the Bishop is vulnerable to space-gaining queenside maneuvers via
- This move does not provide any immediate threats, so Black is free to continue to choose a system as he pleases.
Looking at these two lists, a picture begins to form of the themes for both sides.
White will likely try to hold onto this Bishop, to assist in his kingside operations. He will likely play
4.d3 to keep the position closed, and to cement his hold on the light central squares.
Meanwhile, Black may use the Bishop's placement as a convenient way to make his thematic land grabs on the queenside. Black is not compelled to respond in any particular way, though my database suggests that a Dragon-like setup with
...Nf6, ...Nc6, ...g6, ...Bg7 is slightly more popular in this line than the
[FEN "r1bq1rk1/pp2ppbp/2np1np1/2p5/4P3/1BPP1N2/PP3PPP/RNBQ1RK1 w - -"]
1. e4 c5 2. Nf3 d6 3. Bc4 Nf6 4. d3 Nc6 5. c3 g6 6. O-O Bg7 7. Bb3 O-O
This third move choice from White, while unusual, does not seem to be downright inferior. It suggests a tense, positional midgame, with each side operating on his better flank. Good luck!