One of the things that feels really absent from indie games is intro sequences (though a few do include them). While these days they are generally done with FMV, there was a lot of cool little animated sequences that would play on various arcade machines when left idle for a bit, and there were quite a few on the NES, SNES, Sega Genesis, etc. too.
If a game's got one then generally I'll watch it. A good intro sequence gives you a taste of the adventure you are about to undertake and the thematic tone of the game and can really get the game started off with a bang (by far my favorites being from the Onimusha games, especially 3 which is just insanely awesome). I think if more indie games included intro sequences I'd be more likely to dig in (some better menu presentation wouldn't hurt either).
An added "attract mode
" could go a long way as well since they often give you a taste of what to expect in later levels (which can be helpful if the earlier levels aren't as action-packed, etc.) as well as giving you a rough idea of how the game plays, sounds, and looks overall. In any case, here's a few intro sequences that have really stood out to me over the years:
Onimusha 2: Samurai's Destiny
Onimusha 3: Demon Siege
Onimusha: Dawn of Dreams
Blood Will Tell: Tezuka Osamu's Dororo
Legacy of Kain: Soul Reaver
Final Fantasy 8
And last, but not least, my favorite indie game intro:
I remember one of the Jumper games having a really neat intro too but I couldn't find a clip of it.