A portrait of your father would be the most basic idea but also the most difficult tattoo to produce. Portraits generally need plenty of space in order to show adequate detail and they absolutely require a skilled artist. They can be done in the classic black and gray style or color if desired. Always research an artist's portfolio before trusting him to tattoo a life-like portrait.
A text tattoo of sage advice given you by your father can be a loving idea. There are literally hundreds of styles and fonts to choose from when selecting a text, however an even more personal way to do it would be to have your father write down the advice on a piece of paper and have that image transferred onto you. Just be sure he writes legibly. It can be just a few words or several sentences, it just comes down to how much space you are willing to dedicate to the cause.
Choose a childhood object that only you and your father would know to be significant. It can be any random object so long as its true meaning is shared by the both of you. If you wish, you can lower the rate of suspicion and countless inquiries from others by tattooing the word "daddy" underneath or above it. Again, the script can be done in any style you wish and main piece can be color or black and gray to show nostalgia.
The smallest and most unobtrusive tattoos can often have the most impact. Simply tattoo your father's initials, his birth date or both can be an easy way to get a tattoo without advertising to the would that you have one. Seemingly random strings of letters and numbers can be the most meaningful and significant, plus if your father isn't the type to wish a tattoo on his daughter, this should be a healthy compromise.