Natasha can be a diminutive of Natalia, so the most direct route to use it as an honoring name without using the name itself (or a directly shortened version, like Tasha) would be Natalia or Natalie - even simply Talia would be a reasonable extension of that connection. Going the other direction, Sasha can be a diminutive form of Natasha.
Nadia has a similar vibe and sound and means "hope." Asha ("AH-shuh") also means "hope" or "wish."
Masha comes from Maria / Mary and those names unfortunately are kind of widely "known" to mean bitter / sea of bitterness (they have mixed/disputed origins), but it's another fine name with a similar sound to Natasha.
Hope that helps!