Every Exchange user’s mailbox can have many email addresses. Can a user define what email address to appear in the “From” field?
No, the user can't control it. The secondary SMTP addresses assigned to a mailbox receive email but cannot send email from the Exchange mailbox. If the user needs to send from a secondary address there are several methods they can use to send email from the desired address.
Option 1: Set up a fake POP account (or IMAP) to use for sending. With this option you'll use "nomail" for the incoming server name, and use valid settings for the Outgoing server (SMTP).
You'll need to deselect the option to test the account before exiting the Account Setup dialog, since Outlook will fail trying to connect to the incoming server.
Next, you'll configure the Send and Receive settings so this account is not checked for incoming mail, to prevent annoying error messages from Outlook when it can't connect to the Incoming server. Open the send and receive dialog using Ctrl+Alt+S, click Edit then select the fake account and uncheck the option to check for new mail. To use, select this account from the account list.
Option 2: Create a distribution group and set the user as the only member and give them Send as rights. (A contact in the GAL or a mail-enabled public folder could substitute for the distribution group.) To use, type the address in the From field or select it from the GAL.
Options 3: The administrator can install a third party utility, such as Choose From.