In defense of those episodes, the Beast in the pit wasn’t supposed to be the Satan of Christianity per se, but it was claiming to be the original being that all cultures’ “Satan” myths were based upon, and that had existed, as it put it, “since before Time began.” (We never do find out the truth, it could have just been some big old monster lying to the Doctor to get inside his head.) I’m an atheist & I had absolutely no issue with that storyline. I actually really enjoyed that two-parter as it focused very much on the Doctor & what he thinks and feels.
If I was going to show my dad an episode of Doctor Who it would be “Blink.” I think it could be a positive thing that the Doctor isn’t in it much, since it means you don’t need any knowledge of the series at all, and the Doctor comes across very well in the scenes he is in. His line, “It’s my timey-wimey detector. It goes ‘ding’ when there’s stuff,” sums up everything wonderful about Ten for me. It might make him eager to see more of him.
“Girl in the Fireplace” would be a close second, although the coda at the end (showing the name of the ship) is lame. It’s an incredible episode of television.
TEC/DD is a fabulous 2-parter, but I don’t remember it well enough to know if it would make a good introduction. It seems like it was a major development point in the Rose/Doctor relationship, and it might not be quite as powerful for someone with no previous exposure to the two characters.
If you want a great 2-parter, “Human Nature/The Family of Blood” might be good. You don’t really need to know Martha’s background with the Doctor (largely because it hadn’t been developed that well previously) to get what happens in the episodes. It has possibly the best performances of Doctor Who ever. Every actor really shines incredibly. And it showcases one of the trademarks of Doctor Who - the show’s ability to take faintly ridiculous and contrived sci-fi gimmicks (a fob-watch that contains your memory, an alien monster that can “smell” you across time and space, an army of animated scarecrows) and weave them into something wonderful and fantastic.
I’d have to go with either The Girl in the Fireplace or if you could get him for a 2 parter then Human Nature/Family of Blood.
I grew up in the 70s so Tom Baker is the only doctor for me, of his episodee I would choose: Terror of Fang Rock, Robots of Death and Genesis of the Daleks.
I’ll put in my vote for TEC/TDD, because that was the one that really turned me on to the Doctor, even though I’m not a big horror fan. “Everybody lives, Rose. I could use more days like this!”
Second place mention to TGITF, and a dark horse nomination for the Christmas Invasion.
After pondering the question for a bit I’d go with the first Eccleston episode, since that was what drew me in as well. It has snappy dialogue, funky aliens, great Rose/Doctor interaction, and lots of running. And a great soundtrack. TEC/TDD is also a good choice, but might have a bigger impact later, when (if) your friend knows the Rose/Doctor dynamic.
I’d not go with Human Nature/The Family of Blood since I thought one of the reasons that made the episode so great was that you could tell the difference between “regular” Ten and John Smith; and somebody wouldn’t really be able to experience that if they didn’t know the Doctor at all.
My first reaction was to say Blink, though, since that episode has an appeal transcending Sci-Fi. Which is also the reason NOT to show it, since it’s so atypical.
Blink - only Doctor Who episode to frighten me ever. The writing is incredibly solid, and the fact that the Doctor isn’t in it much isn’t a drawback if you’re trying to draw in new viewers because the focus is on the story and not characters you need to be familiar with. After Blink I’d just start at Rose and go chronologically.
And The Unicorn and The Wasp was horrible. I know at least one person who saw that as her first Doctor Who episode and it put her off the whole series.
Yes, Rose is good too, and the one with Queen Victoria and the werewolf, and the first one with Eccleston and a Dalek, and ‘The Long game’…
Boy, so much good TV, huh? 
I’m stunned you would say the end is lame. It explains of the behavior of the “robots”. Without it, it’s a bit of a “WTF”.
There was also an on-line “TARDISODE” which prefaced the story, showing the
original crew and the disaster that started the process. I was rather annoyed that this wasn’t included on the boxset.