In Final Fantasy VII is there any way to save Aeris?

No. You can cheat with things like the game genie, but she cannot be "legally" brought back.

It is possible to prevent her death; don't progress in the main storyline far enough to cause her to die. This will prevent you from completing the main story, but you will have saved her life! Is finishing the game really worth Aeris' life?

No, but you can see her again. On disc three, revisit the church in Midgar in the Sector 5 slums. A silhouette of Aeris (or her "ghost" as some fans will say) will appear near the flower patch for a brief moment.

Aside from that, only cheats (gameshark, etc.) can get her back to your party.