For Skype on the iPhone (and apparently on the iPad?), there is no way to change an existing contact's phone number. The workaround seems to be to add a new contact for that person with the new phone number. Then get that contact to change his/her phone number in Skype (right?)
As an aside, on my Windows PC Skype app, it took me a while, but I found I could edit the phone number there, by clicking the little arrow head to the right of the "Call Mobile" button, hovering on the Mobile number, and then clicking on the pencil symbol to its right, which is used either to edit the number, or delete it.
This information is no doubt elsewhere on the Forum, much as the needle is somewhere in the proverbial haystack.