I don't have a cheap meter suggestion for you, but I did try a few of mine. Are you 'continuity scanning' where you run the probe down the pins of a fine-pitch SMD package looking for a beep? Or are you just hand-probing one at a time and find some meters very slow?
All of my bench meters that have continuity were pretty sluggish and I can see how a quick fingered person might become impatient.
The only unit I tested with a very quick response and latching (so you get at least a short beep instead of a click or chirp) was my old Fluke Scopemeter with a reliable 1.75mS pulsewidth response. My BK Precision handheld was less than 10mS, but was a faint click or chirp that got stronger up to 20mS or so where it was easily audible even with some background noise. It works fine in normal use, so I would guess that 20mS is fast enough for one-at-a-time poking, but for the pin-scan method, you want all the speed possible. To give you an example of what the numbers mean, with the Scopemeter I could not strike the terminals fast enough to NOT get a beep, no matter how hard I tried, even flicking them past one another so they just tapped as they flew by.
I hope that helps a little. FWIW, I find that the biggest obstacle to continuity checking is getting a reliable connection, not speed. This is especially true on 'old' boards where I typically poke and prod and wiggle 3 or 4X and still occasionally miss a point.