Bounce Servo with 555 timer

Does this do what you wanted (sorry the video is a bit blurry)?

Here's the circuit and the build on a breadboard

the start position is controlled by the 1k resistor under the 1N4148, the end position by the 470n capacitor
controlled by the MOSFET (I actually used two 470n in parallel, so the circuit value's wrong), and the delay time
after the switch is released is set by the 150k and the 10u.

These are the output waveforms going to the servo, in case you're interested. First is the rest position, the second
the position when the switch is pressed.

Anyway, it will give you something to play with if you're interested in experimenting with a 555.

So, full disclosure, I inherited several pounds of components and am just getting my feet wet.  I think I saw some of those orange\black (diode?) pieces.  And I think another video labeled that  black 'D' shape a transistor.(EDIT: that is the mosfet. I had only seen them as rectangles)  I'll have to search for any mosfets.  I don't have a fancy power supply(yet) that I can dial in the voltage...so I'm just playing on a worn out 9v.  I think some math still works as I saw the 555 timer works with fractions of the power (1/3, 2/3).  THANKS for understanding my request!  Seems like this is an answer, but I want to attempt it first.  My personal attempt didn't go great, but I feel this is a working solution and have marked it as such. Thanks for all the quick replies and interest!

A worn out 9V battery won't work too well with a servo. You'd do better with 4 fresh AA batteries for a fairly robust 6V. A low cost way to get the battery holder might be to look for an old consumer device and use the battery compartment (an old wrecked toy or radio, or something like that).

Almost any small n-channel MOSFET would work here. I used the BS170 because I had some, but mentioned the 2N7000 because they're more common and people are more likely to have them. For the connections, look in the datasheet. Beware, though: the connections vary, so you do need to look them up - don't assume they're all the same. Here are the pinouts for the BS170 and 2N7000 and you can see that they're different.

The circuit isn't very precise and the timing will vary as the supply voltage changes. I haven't helped by putting in the diodes to separate the charge and discharge path to get a short pulse time. For the diodes, 1N914 are common too and you may have some of those.

You can extend the delay time by increasing either the 150k or the 10u next to it. If you go too far though, the return of the servo at the end will get more and more sluggish.

One small problem with a servo is that it has a motor and can produce lots of electrical noise. That might affect the timing circuit. If you have any doubts, run the servo on a different supply (with the grounds still connected together) and see if that cures it. If it does, you'll need to clean up the supply to the timing circuit.

A worn out 9V battery won't work too well with a servo. You'd do better with 4 fresh AA batteries for a fairly robust 6V. A low cost way to get the battery holder might be to look for an old consumer device and use the battery compartment (an old wrecked toy or radio, or something like that).

Almost any small n-channel MOSFET would work here. I used the BS170 because I had some, but mentioned the 2N7000 because they're more common and people are more likely to have them. For the connections, look in the datasheet. Beware, though: the connections vary, so you do need to look them up - don't assume they're all the same. Here are the pinouts for the BS170 and 2N7000 and you can see that they're different.

The circuit isn't very precise and the timing will vary as the supply voltage changes. I haven't helped by putting in the diodes to separate the charge and discharge path to get a short pulse time. For the diodes, 1N914 are common too and you may have some of those.

You can extend the delay time by increasing either the 150k or the 10u next to it. If you go too far though, the return of the servo at the end will get more and more sluggish.

One small problem with a servo is that it has a motor and can produce lots of electrical noise. That might affect the timing circuit. If you have any doubts, run the servo on a different supply (with the grounds still connected together) and see if that cures it. If it does, you'll need to clean up the supply to the timing circuit.