As pointed out in the answer to this question, a half-elf can choose the favored class bonus from the typical +1 hp or +1 skill point, or take any of the racial favored class bonuses offered to a half-elf. In this case, focused on bard, the options are:

Add +1 to the bard’s total number of bardic performance rounds per day.

for Half-elf,

Add one spell known from the bard spell list. This spell must be at least one level below the highest spell level the bard can cast.

for human, or

Add +1 to the elf’s CMD when resisting a disarm or sunder attempt.

for elf.

My question is, could a half-elf (or anyone with multiple options like this) take a different one on each level up?

For example, could a half-elf bard take extra health for the first 2 level ups, then take an extra round of bardic performance (the Half-elf FCB) for the next level up, and then switch to taking the extra spells known (the Human FCB) for the remaining levels?

Yes

A creature that has available multiple different favored class bonuses for the same class can pick a different favored class bonus each time the creature gains a level in its favored class. Favored Class, in part, says

The choice of favored class cannot be changed once the character is created, and the choice of gaining a hit point or a skill rank each time a character gains a level (including his first level) cannot be changed once made for a particular level.

The Advanced Player's Guide on Racial Favored Classes—thankfully—makes this a bit clearer, in part saying:

The normal benefit of having a favored class is simple and effective: your character gains one extra hit point or one extra skill rank each time she gains a level in that class (or in either of two classes, if she is a half-elf).… In most cases, these benefits are gained on a level-by-level basis—your character gains the specified incremental benefit each time she gains a level. Unless otherwise noted, these benefits always stack with themselves.… In some cases this benefit may eventually hit a fixed numerical limit, after which selecting that favored class benefit has no effect. Of course, you can still select the bonus hit point or skill rank as your favored class benefit, so there is always a reward for sticking with a favored class.

Until the emphasized text, an assumption could be made either way, but that emphasized text seals it: if a creature has multiple options for its favored class bonus, a different (or the same) favored class bonus can be taken whenever a new level in the favored class is gained.