I try to convert them right back. They don't like it.

It's perfectly reasonable to explain to the person that what they're doing is harassment and they need to stop.
If they persist, it is then reasonable to report them to whatever law enforcement representatives available. On a plane? Report them to the flight attendant.
No one willing to do anything? Turn the tables on them and start pointing out ALL the fallacies connected to their beliefs and their book.

I'd say to her...

"If I was a Muslim, would you appreciate me trying to convert you to Islam?"

Yep. Yesterday at work.......sigh. I stated my belief and then listened to her every 'the lord' and 'the church' she had to offer me after that. Poor thing. I felt sorry for her. She was actually going through a rough time and 'the lord' was how she coped with it. Their everywhere!!!!!!

This is why they sell alcohol on flights.

i had a young man sitting next to me on a plane say: What do you know about the Mormon. church? i could tell he was about to start telling me so i firmly said enough! This answer plus my tone of voice was enough to discourage him.

Ask them to stop talking to you. If they continue, it's harassment. that point, tell the attendant you're being harassed. Then they'll move her. If she resists, they'll ground her

Tell them you are not interested and, if they can not switch subjects, please stop talking to you.

I would put on a highly hostile expression, and tell someone - in my most menacing stage voice (and I've got some good ones!), that if they don't stop, or immediately change the subject, I will let my inner demon surface and put a death spell on them !

Open the door and show her out...

tell her you think she's sexy

I start talking in a very broken English with a very heavy French accent then proceed to repeat their words back to them with the best warped profanity I can find during the moment. "wat do yoo mean Madamme, the playn is jwerking off? I don't underrrr sstane?"

Act like your possessed by a demon!

"Ich verstehe nicht!"

Things that I've used successfully to shut down tenacious thumpers: "Sorry, I don't DO Jesus", "It's against my religious beliefs to support religion" (this one leaves them baffled). "I practice the more traditional way, from before Christians stole it (usually in reference holidays). The key is to be very smiley and friendly, and sometime I'll just pretend I have no idea what they're talking about- this is great for introducing the concept that their beliefs are not the universal default.

On the few occasions that i fly, I bring my MP3 player with me. Funny, when my earbuds are in, no one bothers me.

I would start teaching her the Periodic Table.

I have an app (offline) on my phone. It'll work perfectly on a plane. =D

Ok don't do this - the air marshalls might come after you.

I do think saying "I'm not interested in this conversation." might be the first step.
After that it could get less polite.

Laugh then pull out a smut-filled romance novel to read to them.

Pass gas

I think I would try to spontaneously combust, but I would only be able to do that once.

Seriously, I’m not sure because I’m generally polite as the dickens, but I think I’d politely say that I have no religious faith, and tell them there was nothing they could say that would be meaningful to me.

I’ve had experiences where people accept this and walk away, and others have gotten a little cranky.

Ask the flight attendant for a para and jump out. lol Serious answers, ask if you can be moved, or try converting her to athiesm!

EDIT: Damnit, people already beat me to the point. Oh well..

The first request that she not continue with the subject would be polite but blunt.
Things could then rapidly escalate.

Take a nap.

When it's relevant, I can still pull out the Southern accent. "Ma'am, Ah was raised to believe that discussin' religion with anyone outside a church is rude. So, Ah jes don't do it. Evah!"