Have a look at the AutoPopDelay, InitialDelay and ReshowDelay properties of the ToolTip class, as they control the behaviour of the tooltip.
I generally play around with the values until I get something that 'feels' right. It's annoying when a tooltip shows immediately, and for short tooltips, it's also annoying when they disappear too soon. For really long tooltips, say, several paragraphs (yes, poor design decision, but if there's a lot of info to read, at least let me read it!) then it should stay open as long as my mouse is stationary.
A tooltip example from MSDN gives the following values:
AutoPopDelay = 5000;
InitialDelay = 1000;
ReshowDelay = 500;
// Force the ToolTip text to be displayed whether or not the form is active.
ShowAlways = true;
As mentioned in a comment, the poster wants to trigger the tooltip programatically. For that, ToolTip.Show( ) needs to be called. To get a delay effect, you'll likely want to have a timer running which counts the time that the mouse is stationary. Whenever the mouse enters, leaves or moves within the control, this time should be reset.