If you not see New Meeting Request button in the Home page, make sure Calendar component is added to your Home page layout. After that also make sure Show Requested Meetings in the Calendar Section on the Home Tab is selected under Setup | Customize | Activities | Activity Settings. When this is selected, you will see Scheduled Meetings and Requested Meetings tab under Calendar.
"Show Requested Meetings in the Calendar Section on the Home Tab" NOT selected
"Show Requested Meetings in the Calendar Section on the Home Tab" is selected
For New Meeting Request button in Open Activities related list of Lead and Contact, make sure the button New Meeting Request is added to the page layout.
When you use Cloud Scheduler to request a meeting with a contact, a lead, a person account, or a coworker, Salesforce creates a unique Web page for your meeting that displays the proposed meeting times. When invitees visit the page, they select the times that work for them and send you a reply. Salesforce tracks all the responses so that you can pick the best time to meet when you confirm the meeting.
- You must have at least read access to the contacts, leads, and person accounts that you request a meeting with.
- Each invitee must have an email address specified in their record so Salesforce can send them a meeting request.
- You can invite up to 50 people to a meeting. Invitees can consist of only contacts, leads, or person accounts, only coworkers who use Salesforce, or a combination.
- You can propose up to five meeting times by picking times in the calendar.
- Recommend you to install Salesforce for Outlook so you can sync your calendar events between Salesforce and Microsoft® Outlook®
- Your meeting’s Web page displays all messages between you and the invitees. This includes messages that invitees write when responding to your request.
- If you requested the meeting from an Open Activities related list, the meeting appears on the detail page for that contact or lead only. It doesn’t appear on the Open Activities related list for any other invitee. For example, if you requested a meeting with Jane Smith from her contact detail page and then clicked the To field to invite John Doe, the requested meeting only appears on the Open Activities related list for Jane Smith and not John Doe. If you requested the meeting from the Calendar section on the Home tab, the meeting doesn't appear on the Open Activities related list for either Jane Smith or John Doe.
Currently Meeting Request only available for weekday, vote this idea to include weekend in the meeting request https://success.salesforce.com/ideaView?id=08730000000jA8pAAE and this idea https://success.salesforce.com/ideaView?id=08730000000I4z9AAC
Reference: How Cloud Scheduler Works in Salesforce Classic