It's a bit of a broad stroke, but you can use Trusted IP ranges if the user is coming into Salesforce from a known static IP address. I say broad here as it won't be specific to that user. It will include any other users logging in from those IP addresses. See Set Trusted IP Ranges for Your Organization
A variation on Trusted IP ranges is to Restrict the Login IP Addresses for the Profile. This will remove the need for the verification code, but will also lock that Profile to a range of IP addresses.
Another option I can think of would be to utilise the the security token. Calls to the APIs, say the Partner API login() method, that include the security token appended to the password don't need to go through the verification check. You could establish the valid session using the API call and then bounce the user directly into the Web UI through the frontdoor.
I haven't tried it, but you might be able to use SAML Single Sign-On to bypass the verification step.
EDIT: SAML SSO bypasses device activation (via Gorav)
You would need to figure out if the effort of any of these approaches is worth it versus just doing the verification process when you switch to a new device.