While we normally understand a wit to be someone whose cleverness makes the person amusing, a wit can also be just a very clever, intelligent, or perceptive person in general (Random House Dictionary). We are later given evidence that her cleverness and keen perception also make Viola amusing, but in only the second scene of Act 1, we are only shown how clever she is. We even see her proclaim her own cleverness or wit in the lines, "What else may hap to time I will commit; / Only shape thou silence to my wit" (I.ii.63-64).
In this scene, she has just landed in Illyria, and with both her father and brother now both being dead, she feels uncomfortable confessing to the strangers of Illyria that she is a noble woman and all alone. It would be very easy to take advantage of a woman in her situation. When she first hears of Olivia, she wishes she could work for Olivia until she feels it is safe to disclose that she is a wealthy noble woman. But upon hearing that Olivia will reject her service, she decides she could be of use to Duke Orsino. But again, she wants to protect her true, vulnerable identity and so devices a plan to masquerade as a Eunich and ask for employment. She asks the sea captain to help her conceal her feminine identity, and is so pleased with her plan to protect herself that she calls her plan and herself witty or clever.