Left-back Cole will be making his 102nd appearance for his country and has been handed the captaincy in recognition of reaching his century of caps.
Lampard, who would normally have been captain in the absence of Steven Gerrard, said: "The way Ashley has performed over those 100 caps, he is the most consistent left-back probably in world football over such a long period of time.
"It's the perfect gesture and very much deserved from England.
"I have been lucky enough to play with him for both club and country and with the competition he has had - you talk about Leighton Baines in the last few years, what performances he has put in - and Ashley still remains very much the first person on the sheet when he is fit.
"So fair play to him for that. There are not many players who get to the 100 and do that."
Both Cole and Lampard enjoyed a strong end to the season as Chelsea beat Benfica in the Europa League final and finished third in the Premier League and Lampard is hoping to continue his fine form.
The midfielder, who became Chelsea's all-time leading goalscorer earlier this month, said: "It's nice to come in on a run of confidence personally and as a team, to finish how I did and for us as a club to win the Europa League brings a few players into the squad on a confident run.
"So if we can bring that into the squad, around the place and on the pitch, then it is going to be a bonus."
Lampard is expecting a tough test from the Republic of Ireland in their first meeting for 18 years.
He said: "The type of players that they have, a lot of them play in the Premier League so that adds to the competitiveness of the game.
"I have a lot of respect for the Republic of Ireland. Every game I see them play, the organisation, their desire to get results and they are very proud to play for their country. We have to show the same and try and trust in our ability to come out on top."
Roy Hodgson had initially appeared to reveal that Lampard would be skipper for the game, with Cole leading the team out at Wembley, only for the matter to be clarified later with the left-back confirmed as captain.
The England manager told a press conference: "Frank has always been the captain in Steve's absence.
"I appointed him vice-captain. But in actual fact tomorrow I have discussed this and Frank is happy for Ashley to lead the team onto the field in celebration of his 100th cap which he'll receive tomorrow.
"It's actually his 102nd appearance but he'll receive the cap tomorrow; so Frank and I are happy that he will lead the team out there. But Frank still remains captain of the team - I'm afraid that's a bit more complicated than you maybe would have liked it to be, but that's the way it is."
Hodgson then said that Cole would in fact "wear the armband" in the friendly against Giovanni Trapattoni's side - a sign, of course, he would be captain.
"After speaking to Frank and sounding out the other members of the squad... everyone thought it was would be nice if he (Cole) had the armband on and led the team out," Hodgson added.
"Frank was happy to hand over that honour because we all believe he deserves it; he's been a fantastic servant not only to Chelsea and Arsenal but England too."
What Hodgson appears to have tried to say is that Cole will in fact be captain against the Irish, but that Lampard spoke to the media because the Chelsea defender does not enjoy the attention that comes with speaking to the media on the day before the game - a responsibility usually shouldered by the skipper.
"Perhaps he (Cole) is reluctant to stand in front of the media and play the captain's role," Hodgson said.