If your IP address changes, you reconnect to the network and broadcast a new node_announcement message containing the set of your node's reachable addresses and an updated timestamp. The node's public key identifies the node and all of its channels and may not change. All other information in the node_announcement message can be changed.
You can assign an alias to your node. The als is what is being depicted in node explorers. It is a regular utf8 string. Nowerdays there are a lot of icons as characters in utf8 encoded strings. For example ⚡.
So in lnd and in clightning (apparently both have the same Syntax) you could add the following line to your config file:
alias=⚡my node name with ...