Hi there, Amy! This is Marc. Thanks for reaching out to the community.
First, remove the cached entry start typing the person’s email address in a new message, then when it appears in the dropdown, click the X next to it to delete it. Send them a new test email, and Outlook should pick up the correct name from your updated contact.
If that doesn’t work, check for linked profiles in People view and unlink any that might have the wrong details. On Outlook.com, you can also go to Settings > View all Outlook settings > Mail > Compose and reply > Contacts and clear the suggested contacts cache. Once the incorrect entry is gone from both cache and contacts, new messages should display the right name.
Let me know how it goes.
Best regards,