I hear it all the time VFR at CRQ. In this case what you were requesting is the standard noise abatement procedure, and as such they dont really expect you to ask for it, they just expect you to do it. It's also shown on a little diagram right next to the hold short lines, but I didn't notice that sign until I'd already been to SMO a couple times. If you simply ask for straight out, they expect you to do the dogleg over the golf course and go straight out from there. The other way I've heard it requested is "North at the coast" or for me since I have to climb to 3500 to head south through the LAX SFRA the call is "Right climbing 270 at the coast for Special Flight Rules"
From the context given, and what I've seen many times at Palomar what I get is this. You've made a request, and the controller has gotten it. The clearance for that request generally comes with the clearance for takeoff. However they currently cant clear you for take off because someone is on final, still on the runway, etc. "On Request" basically means "I got it and you'll get the clearance along with your takeoff clearance". It's not quite the same as saying "stand by" which really means something more like "I'm busy at the moment, I'll call you back when I'm ready to take your request"
There are a number of airports that I've found that have their own unique procedures for departure, and they seem to use this phrase more than others. Palomar has the "Alpha North" and "Alpha South" departures, which simply mean climb runway heading to the coast then turn north or south. SMO has that funky dogleg to get over the golf course, then straight to the coast before turning. Both are considered standard procedures, but obviously only for those airports. They cant clear you for an Alpha North out of Palomar until they can clear you to takeoff however, so if someone is on short final you'll get "on request" then "alpha north approved, cleared for takeoff runway 24" when that plane is clear.
Clear as mud?