or 2) I kinda like the idea of applying strain to every failed attempt (maybe even additional strains if Threats are generated). This way, they can attempt as many times as they want before their character is too tired or incapable of continuing and needs rest.
This is okay unless you've got the Heal control upgrade that allows for strain to be healed. 1 light side pip and *poof,* those last 3 or 4 checks' worth of strain are gone.
This is why I am a fan of not limiting it "only 1 check and that's it for the day," or any other houseruled patch, but rather setting a reasonable limit as the GM is supposed to do. Maybe they have 8 hours of downtime, and each skill check they perform represents an hour of work. They can either take a rest and benefit from natural healing, or perform several skill checks with the possibility of losing needed sleep (and now they're going to be making Resilience checks to avoid the effects of sleep deprivation). The possibilities are endless if you just keep it centered in the narrative and don't apply a hard and fast rule in an attempt to "fix" a perceived problem.
Oh yeah, I meant to mention this option too. That even given a lot of time, only *so* many attempts could reasonably be made. So maybe give the player a hard limit up front ("you can try 5 times") or whatever sounds right.