There are two separate issues: What meter should you use, and what probes should you use.
If you're only doing continuity detection, it's hard to imagine a meter that won't do the job. A lot of people, myself included, like the Fluke-style latching continuity detection, because it delivers a good solid audible beep even with a very brief contact. Dave has demonstrated this well in some of his fairly early multimeter shootout videos. Some cheap meters have slow response in continuity mode.
I don't have enough experience with bargain meters in continuity mode to make specific recommendations.
As far as probes go, as long as your meter uses standard shrouded banana jacks, you can use any standard banana probes. There are several makers of good test probes with the standard telcom style "bed of nails" clips on the end. Fluke makes their MT-8203-20, which is one option, but similar products are available from other manufacturers.
For this particular application, I can imagine spending as much on the test leads as on the meter. Great leads combined with a really cheap meter should do the job well, while a superb meter outfitted with standard style leads wouldn't be so great.