Many players love the name they already chose and only want it to look a bit more special. In that case, keep the name and change the presentation. The same five letters can feel completely different in small caps, a gentle script, or a bold style framed by a pair of symbols.
Try a few variations of the exact name and compare them together. Keep the spelling identical so friends still recognise the pet, and avoid swapping letters for look-alike symbols that make the name harder to read or search. Count the characters in the styled version too, because frames and decorations eat into the 20-character limit faster than you expect.
A name font studio lets you style an existing Adopt Me pet name by typing it once and comparing many Unicode font-and-frame recipes in one view. Pick one that matches the pet's theme, copy it, and check it in the game before you save.
Disclosure: the linked Adopt Me Fonts site is one of our own projects.