Does God give the stranger a "better" name because He knows how much more the stranger has to give up to follow Him (Isaiah 56:3-7 Isaiah 56:3-7  Neither let the son of the stranger, that has joined himself to the LORD, speak, saying, The LORD has utterly separated me from his people: neither let the eunuch say, Behold, I am a dry tree.
 For thus said the LORD to the eunuchs that keep my sabbaths, and choose the things that please me, and take hold of my covenant;
 Even to them will I give in my house and within my walls a place and a name better than of sons and of daughters: I will give them an everlasting name, that shall not be cut off.
 Also the sons of the stranger, that join themselves to the LORD, to serve him, and to love the name of the LORD, to be his servants, every one that keeps the sabbath from polluting it, and takes hold of my covenant;
 Even them will I bring to my holy mountain, and make them joyful in my house of prayer: their burnt offerings and their sacrifices shall be accepted on my altar; for my house shall be called an house of prayer for all people.
American King James Version×)?
An excellent example of what the stranger has to give up is given in the book of Ruth. Ruth left her family, her native land and customs—everything that was familiar to her—and went with her mother-in-law to an unknown land to live among people whom she must have suspected would hate her (after all, she was a Moabite woman), all to follow God.
"'Look,' said Naomi, 'your sister-in-law is going back to her people and her gods. Go back with her.' But Ruth replied, 'Don't urge me to leave you or to turn back from you. Where you go I will go, and where you stay I will stay. Your people will be my people and your God my God'" (Ruth 1:15-16 Ruth 1:15-16  And she said, Behold, your sister in law is gone back to her people, and to her gods: return you after your sister in law.
 And Ruth said, Entreat me not to leave you, or to return from following after you: for where you go, I will go; and where you lodge, I will lodge: your people shall be my people, and your God my God:
American King James Version×, NIV).
I know everyone focuses on the words, "Where you go I will go," but what about the words, "Your people will be my people and your God my God"? We can see in verse 15 that the other sister-in-law had gone back to "her gods." But Ruth made, at some time, the decision to become a follower of the God of Israel.
Ruth put her faith in God, made her decision to turn her back on the land of Moab and followed a whole new path, wherever that would lead her, despite the hardships or tragedies and despite Naomi's bitterness of the moment.
God valued Ruth's decision and faith, so much so that Ruth, a Moabite woman, married Boaz and was counted in the direct lineage of Jesus Christ.