On my Android 2.3 phone I've exported my contacts and imported them on my new Android 4.4 phone, a Motorola G. Unfortunately, I found no way to edit the contact. When I open the "phone" application from the main screen, I click "All contacts" (translated) to get the list of all contacts. But as soon as I click on one item, it dials. Click-and-hold does nothing - no context menu occurs. How to edit the contacts (name, phone number)?
Don't use the phone/dialer app to edit contacts: use the main People app (it's called Contacts on some non-Nexus phones). From that app, when you click the contact you want, it displays that contact, and you can click the pencil icon on the action bar (at the top) to edit the contact. (The icon might look different on your phone, because most manufacturers replace the People app with their own variant.)
I just did this on my Nexus 4 w/4.4.2 and the above answer is partially right. Open your contacts and tap on the picture to open up the contact. Now tap the silhouette of the person next to the star. Now you can tap the 3 dots in the upper right corner and that opens up the editor. Finish the edit and tap done in the upper left.