The presentation that senior Lego designer Jamie Berard held at Brickcon 2006 and that spawned the entire legal vs illegal discussion can be found here:
The particular case you ask for is discussed in slides 9 and 10:
A 1x1 cone with an inverted positioned screwdriver just might work here.
So, I'm saying just might based off the the halved cast line of Peggy's head.
The base peg of the horn/drill-bit would be too wide for the hole, but based off the cast fault line, which I hi-lighted in blue, there is a greater chance for the plastic halves to ease the slimmer screw ...
Put two plates between each layer of Technic bricks and the holes will align.
This technique was used commonly in studded Technic sets. Here's a page from the instructions for 8042 where you can see this used in several different places:
If the problem is that the base of the horn is too big to fit in the hole on the chanfron you have to make the former smaller or the latter bigger. Careful work with a file/scalpel, if you're good at that kind of thing. Remember that it's a lot easier to take a bit more off than to put a bit back.
Odd that the bow doesn't use a standard size stud anyway (...
A possibility might be using Horse Battle Helmet (unicorn) parts 89524 or 13745:
However I'm unsure if those would fit. The head geometry of the horse part in question (93083c01pb08) seems to be wider than the geometry of the horses in the castle sets (e.g. 10352c01pb03)
If you're not afraid to modify (cut and/or bend) one of the horse helmets, they might ...
According to LEGO, this type of connection is illegal, since the pin is stressed when connected to an anti-stud and eventually deforms.
People's MOCs do not follow same policy as TLG does. So you may encounter illegal connections from time to time. Pins are also cheap and usually owned in high numbers, so nobody's really bothered if one gets deformed.
I don't have this particular piece but my advice for most 'non-standard' connections such as these is to use a part that is flexible enough to fit the gap.
The 'Technic, Axle Connector Double Flexible' piece under compression can make a snug enough fit to mount a sharks head.
In this example, you can use two Technic Axle-Pin attached to another two ...
In general, this sort of connection is illegal, but I think that it works fine with a Travis brick. Technic pins must be "in click" in a model, or the pins will be held in compression and can be damaged over time.
Here's a slide from Jamie Barard's classic "Stressing the Elements" presentation that explain the need to have Technic pins in click:
And here's ...
I don't think there are any pieces LEGO intends for you to attach to the side of these axle shaped pieces. However, I'm pretty sure you could attach piece 32172 to some extent (assuming the piece being attached to is actually the right size for an axle):
It probably would not connect very well, only the clips on the sides of the socket would hold it, so it ...
this way it can open and close
Blue = 4276 Hinge Plate 1 x 2 with 2 Fingers
Yellow = 4275 Hinge Plate 1 x 2 with 3 Fingers
LightBlue = Plate 2 x 4
White = Plate 2 x 2
Pink = Plate 1 x 4
Light Bluish Gray = Plate 1 x 3
Yellowish Green = Plate 1 x 2
Red = Tile 2 x 2
Green = Tile 1 x 4
Okay, it seems I indeed made a mistake during building, which I solved afterwards:
When assembling the rear mudguard, I noticed that the rear wheel had quite a bit more variability in its sideways adjustment and was quite a bit off center, specifically too far to the left. This caused the chain belt to either run slightly skewed or have friction on some ...
I am still waiting for the bricklink delivery with 27448 tiles that Alex suggested in one comment of his answer. Therefore, I tried a less subtle solution:
This is the starting point: A DUPLO pig that needs to be put directly on LEGO pieces.
I decided to cut a DUPLO piece to regular LEGO size. A DUPLO 2x2 brick was converted to a ...
... 4x4 tile, ...
There is a bit of a challenge with a geometry. LEGO doesn't produce much symmetrical hinges, otherwise they would be out of system. I could think of the following one piece you could achieve a close result - Hinge Plate 2 x 4 with Pin Hole and 3 Holes - Top. It comes in White color as well, so you could incorporate in your layer of white plates.
Here's a ...
Scientific progress goes OINK!
This is an older DUPLO pig from the 70's, and its different design offers us some new possibilities.
For example there's a rectangular hole below the neck.
Now before we go any further, the LEGO policeman would like to remind everybody that stuffing plates into rectangular holes or jamming them between studs isn't exactly ...
Have you tried horizontal clips? They will be a little taller than 1 plate in the center, but you might be able to adapt them to your needs given that the connection point is just one plate. Perhaps these parts:
Something like 4623 with a vertical clip might also be a possibility depending on your constraints:
Mr. Blue and Ms. Red live next doors. Their houses are simple. The walls are one stud wide, the ceiling's one plate thick, some slopes on the roof.
Now they want to move their houses wall to wall. Mr. Blue bores a hole into the wall and happily bolts in a big peg. But Ms. Red thinks that's just a big eyesore!
So they try to hide some clips in the ceiling, ...
Although technically the connection is illegal, I don't think the pin is under stress. I remember this set from my childhood, and the "rotor" rotated easily, which would not have been the case if the pin was stressed.
It doesn't look like there's currently any way to make this connection by normal means.
While there are quite a few parts with similar 1.6 mm attachment holes as your daughter's horse has (including some mildly surprising ones like this ladybug), parts with the matching pins seem to be relatively few. Basically all of them seem to be small ornaments like ...
You've already been given a number of solutions, where the part that actually goes within the components is only one plate high, while the actual connection is a bit more. I can't do any better, but I want to suggest another option:
A combination of
The right combination of these parts allow some offset (if the parts fit), and these ...
Don't have too much experience with DUPLO, so my suggestion is based on my limited knowledge.
I'm not aware if there a way to attach pig directly to LEGO bricks, however I thought of a workaround. Here, green DUPLO figure legs represent your pig feet.
I'm very late to the game here, but why can't we use the existing hole as a guide for an appropriately sized drill bit, converting it from Friends accessory size to standard LEGO horn size? If you drilled it slowly by hand, it would be really hard to mess it up.
This part (Technic, Link 1 x 16 #2637) appears to be an axle with two connectors on each end. Bricklink classifies this piece as a Technic Link and not an axle because it doesn't perform most of the same functions as an axle.
I believe the decision to use the Technic 'axle' shape for the length of the piece rather than a round piece, is because it functions ...
I believe I may have found a workable solution and the "best" solution based on the criteria:
I still wasn't able to find the rubber Technic connector part I wanted, so I kept looking at other connectors and how they fit together, taking special care to note how much space they would take up.
I looked more closely into the "Pop-Up Book Clasp" option (in ...
PROJECT: Dinosaur mount.
Black plaque construction and cradle.
Mounting plaque to a wall.
Reconstructed Ambo100's nice entry, which works beautifully for
Mr. Shark, but the width is too sore for Lady Dinosaur.