I see no reason not to allow a player to "take 1" if he wants to. But a roll of 1 is simply always a failure (on attacks/saves, they might still succeed on skill checks), there are no extra side effects unless you are using variant rules for critical failures or something. For example, if a player wanted to make it look like he was attacking his opponent (just for show or something), but didn't want to actually his opponent, I would probably allow the player to automatically miss. Waving a sword around, even if you aren't trying to hit can be dangerous though, so I might make the player roll anyway and on a roll of 1 the player accidentally hits, so the player would be well-advised to strike for nonlethal damage if this is the case.
But the "luck" of rolling a natural 1 is generally just a failure. In my campaigns, when a player rolls a 1 on an attack or save, I describe it as an ill-fated attempt of some sort, but it doesn't have any extra bad mojo than a normally failed attack roll or save would have. The only exception I can think of is rolling a 1 on a save versus an area of effect spell, in which case an item on the player's possession might be damaged. If the player really wanted to take a fireball full in the face, I'd have him roll 1d4 to see which item was affected as per the PH for rolling a 1 on a save versus an area of effect.