Cross has been ruled out for the visit of Ospreys with a shoulder injury and, as a result, Meyer, who previously captained Counties-Manukau in New Zealand, has been handed the armband.
Meyer, 29, announced he was surprised by the honour: "I'm one of the new guys, so I'm both very surprised and honored to have been asked to captain the side so early on in the season."
The match against Ospreys enables the former Auckland scrum-half to pit his wits against ex-All Blacks number nine, Justin Marshall and he admits he is excited by the challenge.
He added: "Justin is probably New Zealand's greatest-ever half-back, he played 90-odd Tests for the All Blacks. I've played against him a couple of times.
"But I don't think you can ever go into a game focusing on one player. You just need to go in and do your own job and look after our own team. Now I've got an added responsibility as captain so I've got to make sure I do well."
Edinburgh team: Nick De Luca, Andrew Turnbull, Ben Cairns, John Houston, Andrew Maxwell, Phil Godman, Ben Meyer (capt); Augusto Allori, Andrew Kelly, Geoff Cross, Craig Hamilton, Ben Gissing, Matt Mustchin, Ross Rennie, Roland Reid.
Replacements: Steven Lawrie, Ryan Grant, Fergus Pringle, Scott Newlands, Greig Laidlaw, Alistair Warnock, Matt Dey.
Guinness PRO12
Suite 208, Alexandra House,
The Sweepstakes
Ballsbridge, Dublin 4, Ireland