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He's a very smart guy, who demonstrated in winning a number of memory challenges that he's sharp, and he improved his athletic skills against Shane.
Yes, he trusts people too much, that's for sure, and is clearly a bit naïve, but it's a testament to a man who's honorable and sticks to his word.
Especially since it turned out to be a double eviction night and Ian managed to win the HOH competition, convinced he could screw Frank. To make matters worse, he nominated his household crush, Ashley, who'd been so sweet to him but was straight that their relationship was platonic.
Though he never admitted it, he was probably upset she was more enamored of the hunkier Frank.
Frank started flexing his muscles and won his second Power of Veto to save himself, but was incensed to see Ian needlessly voted to evict Boogie, because Boogie would have been evicted even without his vote. Ian even crowed about it during the videos displayed while Boogie sat with Julie, the kid so proud he'd tricked the master Boogie, who seemed oddly to enjoy the betrayal. Up to that point he could have profited just as easily with his relationship with guys who'd demonstrated their loyalty and faith in him.
Not to mention the jig was up and now Frank knew what Ian had done, but Ian was so full of himself he was convinced Frank was no match for him.
True to family form Willie nominated Frank, who looked a goner after the first week.
But Ian didn't count on Frank pulling the Veto out of the hat again, and then he won HOH. Through it all, Frank had made deals with Britney twice, and twice in the diary room she revealed she would let him believe what he wanted and plotted to get him out.
In fact, during the second such situation, when Frank nominated Dan and Danielle, because he was convinced Dan had turned Ian against him, she was determined to win the power of Veto so that Dan and Danielle would be off the block and their alliance would be safe.
However, Shane is a physical guy and won the Power of Veto, removing himself from nomination.
Frank couldn't understand why there was a campaign to undercut him and made a deal with Shane to work together.