The best replacement for webmail is email.

The best alternatives to webmail are rated as following:
★★★★★  email
★★★★  email client
★★★  pop3
★★  client
★  desktop client
★  outlook express
★  net.mail
★  imap
  client mail
  hosted exchange
  web mail
  desktop mail
  client email
  desktop email
  pop mail
  roundcube

Your alternative for webmail is not listed? Suggest!
Speaking about webmail, one probably shouldn't forget about pgp.

Other search terms related to webmail and their ratings are:
★★★★★ pgp
★★★ hmailserver
★★ thunderbird
You have in mind a companion for webmail and it is not yet here? Propose!



Google Trends

Scale is based on the average worldwide traffic of webmail in all years. Click on graph for more.
webmail
email
email client