Cole is back at the club where he started his career and has signed an 18-month contract after leaving Anfield on a free transfer.
Allardyce said: "It's a good one for us. I think that somebody as talented as Joe and as enthusiastic about playing football and not all about money, as most transfers seem to be about today, is a pleasure to bring back to the club and give the opportunity he is looking for.
"I think that we will give him that and hopefully he will grasp it with both hands and produce some of the outstanding ability and quality that we know he has and do that for us during the period of time that his contact is with us.
"If we can extract the best out of him again then it will be a fantastic signing for us at West Ham.
"He is somebody that is coming back home from where he started. He has realised his dreams here of being a footballer and it's great to have secured the deal and really great that he chose us.
"It's nice to have a player who has played 56 times for England. He's just into his thirties so I think there is an awful lot of football left in him. He hasn't being playing regularly enough since his move to Liverpool which frustrated him.
"He feels he needs a break to get somewhere else and recreate himself and I think we have got the back-up services and the staff behind the scenes to help him do that. For us it is a great move and hopefully it is for him.
"Extra quality in the team is always important. Joe's main strength lies in possession and his ability to open teams up and beat individuals is a key element and that is what we will be looking for."
When asked about the possibility of Cole adding to his tally of 56 England caps, Allardyce added: "I don't see why not.
"If he can recapture his old form and play on a regular basis for us, he is bound to get a bit of interest from Roy Hodgson. I think his main focus first is making sure he plays for West Ham on a regular basis and reproducing the form that we know he can produce."