How can I make a redstone button with an X second delay?

Here's a working concept of what you're probably looking for:

The trick is to split the output with varying delays, then each of them take their turn keeping the gate lifted up. The more rows of repeaters you have (with increasing length), the longer the button's signal.

Another concept to use is a little bit more complicated to set up, but is more efficient and space effective in the long run.

1. Connect your button to an RS NOR latch to trigger it ON when you press the button.
2. Create a long line of repeaters that matches the amount of time you want the button to stay "pressed".
3. Connect your line of repeaters to turn your RS NOR latch OFF.

The result is a button the turns the output on, then after X seconds your repeaters will turn it back off.

If you don't know, an RS NOR latch is a bit that "remembers" it's state. Build it by making a redstone clock with 2 nodes.

"-" = Red stone
"O" = Solid block
"*" = Torch on block

--O
*X*
O--