How to enable ethernet 1 Gigabit advertising or troubleshoot the situation?

The key lines are:

    Link partner advertised link modes:  10baseT/Half 10baseT/Full
                                         100baseT/Half 100baseT/Full

This likely means that the other device (most likely a switch) is not advertising 1Gbps, or that the cable in question isn't capable of supporting 1Gbps. Check both, and verify that the cable has all eight pins connected - if in doubt, switch out the cable.

You can also use mii-tool -v to verify the information from ethtool.

Alternatively, you can use ethtool to force the NIC to 1Gbps, but be aware you may lose network connectivity:

ethtool -S eth0 speed 1000 duplex full autoneg on