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Enable IMAP access in Gmail. The option is accessible from the Settings cog icon in the corner -> Settings -> Forwarding and POP/IMAP. There, select "Enable IMAP" and click "Save Changes" at the bottom of the screen.
Now we're going to make an app password for SMS Backup+. Go to security.google.com and select "App passwords." You'll be prompted to sign in again.

On this screen, there's a pair of drop-down lists for categorizing the app password you're about to make. Select "Other" from the left-most drop-down list and give this password a descriptive name like "SMS Backup+" so you know what it's for later.

Click Generate.

Google will give you a randomly generated sixteen character password. Make a note of it, but please treat this as carefully as your normal password. Don't write it on a post-it and stick it on your monitor or copy it to Google Drive/Keep. It's a password that grants access to your Google account without the security of two-factor authentication, treat it as such.

Now open SMS Backup+ on the phone you'd like to use.

Tap "Advanced settings."
Tap "Custom IMAP server."
Tap "Authentication."
Select "Plain text."

Most of the default settings the app loads for Gmail-based IMAP will be fine, but we'll need to change two. Tap "Username" and enter your Gmail address, hitting "OK" when you are done.

Tap "Password" and enter that app password generated up above in step 5 — no spaces. Tap "OK" when you are done.
Double-check that the setting in the Security is set to "TLS." It should be by default, but you should be sure.
Back out to the app's main screen and backups should now work, though the "Connect" section for linking to a Gmail account will now be grayed out.