Make a custom button in Record Pages only visible for particular users

This feature, previously a request in Idea Exchange: Limit the Visibility of a Custom Button Based on Profile is now generally available for Custom Objects and Beta Standard Objects.

To know more about this feature you can review the following resources:

Create Dynamic Actions in Lightning App Builder

The workaround is also still an option, since the Users are related to a Profile you can use a Formula to restrict ‌access to a specific profile.

A good way to resolve the issue is to create a formula field and take advantage of the HYPERLINK function (add an image to the hyperlink to look like a button) to conditionally display the link. For example, the following formula displays the Google hyperlink to "System Administrators" users only:

IF(CONTAINS($Profile.Name, "System Administrator"), HYPERLINK("http://www.google.com/"​,"Button name")," ")

No other user is able to see the "button" (Image).

NOTE:  Support for buttons and formulas is limited. See the Scope of Support for Buttons article, and the Scope of Support for Formulas article for more information.