What you need to avoid happening is your body discharging static electricity to the circuit board you are going to work on unless, that discharge is directly to the circuit board's ground or 0 volts. If you discharge static to the circuit board ground that action can be regarded (in normal circumstances) as being fairly safe for components on the circuit board.
Ideally though, you need to equalize any static potential on your body with respect to the target circuit gradually to avoid creating a spark or a high energy impulse. This is why 1 MΩ resistors are used in the wrist strap when it is normally connected to the anti-static mat on which your target circuit board sits on.
But, by the sound of it, you don't have the luxury of a mat so you have to "tread" carefully and connect your earth strap (through a 1 MΩ resistor ideally) to a port on your laptop that could be construed as being suitable. By "suitable" I mean closely bonded with the 0 volts on your laptop circuit board.
Connecting your earth strap to ground (or earth) on a wall socket in your house is missing the point unless, your laptop circuit board's 0 volts is also connected to that earthing point.
So, make an equipotential bond via your wrist strap to the laptop circuit board and be very careful in how you choose the electrical point to make that connection. Any other equipment or components you might use should also be carefully bonded to that common point and, this is why ant-static mats are used; they keep everything on your workbench at the same potential.
If you don't have an antistatic mat be very careful and think twice about collecting the equipment and components around you that you need. There's nothing much you can do about signal generators other than electrically bond them to the common point but, components can be kept in your pockets until needed and don't let go of them until they are touching the circuit board.
Better still, get a mat and do it properly.