When ad-libbing off script you generally won't have a lot of time to think before speaking, so it's important to keep firmly in mind your character's status and attitude.
Status is the character's place in society relative to the other characters and the audience. A butler would normally be subservient to his employer, and probably the audience in their position as "guests", but might answer haughtily to the maid whom he feels to be above. A king, on the other hand, might very well get annoyed at the impertinence of the commoner audience at asking him personal questions. It would say a great deal about his character if he is yelling at everyone one minute, but meekly subservient to the queen the next. You should know ahead of time where your character stands in the cast pecking order.
Attitude is the most distinguishing characteristic of your character's personality, their primary defining trait. The butler may answer subserviently, but if he makes faces behind the lord of the manor's back -- that's his attitude. A woman who acts bitchy throughout the play will be just as bitchy with the audience on her own. If the sleazy lounge singer has been set up as a Romeo, coming on to all the women in the play, he'll notice an attractive woman in the audience asking him a question before he answers it. And the pompous old windbag who is always ready with a platitude or cliché will always answer with one first. So, look for those defining characteristics and play them up; speaking with a recognizable attitude is not only a good way to stay in character or stall for time while you think of an answer, it's often a good way to get a laugh, too.
Jack Benny got one of the biggest laughs in his career strictly on attitude. Jack's character trait of cheapness had been well established on his radio show, but we didn't know just how far he would take his unwillingness to part with a buck until, one day, he was accosted by a mugger: "Your money or your life?"
The long pause as he considered his answer brought on audience howls. His final, "I'm thinking," brought applause. No joke, no clever word play, just attitude.
Next Step: Justification.
For an excellent discussion of the role of status in improv comedy, read Keith Johnstone's book
Impro : Improvisation and the Theatre,
one of the classic texts of improv.