Pygame Set Mouse Cursor from Bitmap

Another option is to simply hide the cursor, load any arbitrary bitmap that you like and draw it every frame where the cursor is.

Since @SapphireSun asked in a comment about another way, I'd like to present a different answer, and this is drawing the cursor manually. I guess this is what @Mizipzor suggests in his answer, so maybe this is just an elaboration.

First Hide the mouse-cursor, then every time the screen-frame is updated, "manually" paint the cursor:

pygame.mouse.set_visible(False)  # hide the cursor

# Image for "manual" cursor
MANUAL_CURSOR = pygame.image.load('finger_cursor_16.png').convert_alpha()

# In main loop ~
    ...

    # paint cursor at mouse the current location
    screen.blit( MANUAL_CURSOR, ( pygame.mouse.get_pos() ) ) 

This method allows the PyGame program to have any sort of bitmap for the cursor. There may be some trickiness around getting the "click" hot-spot in the right location, but this could be achieved by setting a transparent cursor, with the hot-spot at the position to match the custom bitmap. See the manual for details.

# transparent 8x8 cursor with the hot-spot at (4,4)
pygame.mouse.set_cursor((8,8),(4,4),(0,0,0,0,0,0,0,0),(0,0,0,0,0,0,0,0))

I am not sure how I feel about multi-coloured cursors, but at least it's possible.

You can load cursors in PyGame with pygame.cursors.load_xbm -- that's black and white cursors only, of course, a well-documented limitation of PyGame, per its docs (and I quote):

Pygame only supports black and white cursors for the system.

For XBM format docs, see e.g. here. You can prep such files e.g. with the PIL library, per its docs.