I want to replace the class based saves for a variety of reasons:
1) To have the various saving throw bonuses of characters less far apart and prevent automatic success and failures for level appropriate threats.
2) I want to discourage multi-classing for dip benefits, not multi-classing in general
3) I see no reason to maintain it, ability scores and potential feats do a well enough job to explain why someone is more likely to succeed or not
4) I wish to equalize the classes a bit more, it seems that the weaker classes often have the weakest saves as well (rogue, fighter), or would typically lack the ability scores to boost saves (wizard, sorcerer), while other classes with already strong saves have them enhanced even further with high ability modifiers (cleric, druid, paladin)
5) I like the correlation with a creature's / character's deductible traits having a relation with his strong saves rather than meta gaming
6) it's meant for a relatively low magic campaign and the characters might miss out on the assumed bonuses inherent in magical items, so slightly increasing the saves overall might be a good thing.
The base save will be a bonus of 1/2 your level in every category modified for ability modifiers.
At level 5, 10 and 15 you can pick to get a +1 bonus on one of the various saves, at level 10 and 15 you can pick a save you did not choose to increase before, your other previously chosen increases go up +1 as well.
At level 20 they all go up by an additional +1.
So at level 5 you can choose to increase fortitude, at level 10 you pick reflex and increase your fortitude up by +1 as well. Your saves are now +2 fortitude, +1 reflex. At level 15 your will save gets +1 and the other saves go up +1 as well. Your saves are +3 fortitude, +2 reflex and +1 will.
When your fortitude goes up by +1 you also get a bonus of 5 hit points per increase.
When your reflex save goes up by +1 you also get a +1 bonus on initiative checks per increase.
Nice. Pretty well thought out. I have been considering adopting the Star Wars Saga Edition saves, but with the 1/2 level base as you've done. Classes get a Class bonus to various save categories. So, for instance, I'd give Fighters +2 Fort, and +1 Ref. Rogues probably +1 Fort, and +2 Ref. Clerics would have only +1 to Fort and +2 to Will, etc. These Class bonuses overlap, but do not stack.
The thing that is missing is Prestige Class boosts to saves, in PF we pretty much don't use them anymore with the myriad archetypes and class powers all the way to 20. In Saga, it is expected that you'll Prestige. In that system, the PrC gave larger Class bonuses to saves, again not stacking but overlapping. This buffed out your saves a bit at high level.
I do like your bonus choice progression. If you took Ciaran's concern to heart, you could institute a Saga type base set, and allow choices at later levels. Or boost all saves every so many levels. Just thoughts.
EDIT: Just occurred to me, the smoothest way to address that concern, would be to simply start your bonus choice progression at 1st level instead.