Extra Discovery, hands down. While you will have ten more bombs per day as a gnome, the versatility of extra discoveries is far more valuable than the extra bombs. You can always spend character feats to get more bombs or discoveries.
You just need to decided if you want immediate gratification or delayed gratification here.
For pure stats, goblin would be best with their +4 dex, but getting 3 Extra Discovery without actually having to use up a feat slot sounds pretty good.
Gnome also has the pyromaniac racial option to count as 1 level higher for bomb damage. Higher dex is good, but bombs are already pretty accurate as touch attacks.
i love ratfolk alchemists. Cheek pouches and cleanliness alt racial traits, a tail blade to always be armed, perfect stat bonuses, extra discoveries, excellent skill bonuses, all delicious.
Goblin having access to winged marauder is tempting, however.
INT gives extra bombs, sets the saving throw for the bombs and increases the damage to their bombs. It also gives them more extracts and boosts the saves on any extracts that have a save. If you are looking best race for an alchemist, they should have a boost to both DEX and INT.
That brings the choice down to Ratfolk and a small Tiefling. Tinker gives a bonus to Craft Alchemy and UMD. UMD is a class skill for alchemists and the trait pragmatic activator can make it an INT based skill. This could be very useful for the character as it could open up using wands as well as bombs in combat. A bonus to craft alchemy is also a good for an alchemist. Swarming can be exchanged for Cheek Pouches. Being able to carry 10 pounds of potions/extracts/bombs in your cheeks seem fairly good.
Skilled gives a bonus to bluff and stealth. The bonus to stealth is great but bluff does not really do much. Fiendish Sorcery is useless for this character but can be traded for Prehensile Tail. You can also trade away Darkness for Maw or Claw to get a natural attack.
My recommendation would be to go with Ratfolk. They get more boost in alchemy related abilities than the Teifling. Ratfolk seems like it will make a more versatile character. The Extra Discoveries give you more options with your bombs. A high UMD gives you the ability to use magic items of other classes. Wands are the obvious combat use, but using scrolls lets you actually cast spells, not just use extracts. Versatility is power.
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Might I offer you a fifth option? (Assuming your GM allows it.)
Wyrwoods are a small race with ability score bonuses to Dex and Int, 30’ speed, and the construct type. They have an alternate racial trait that makes their Int count as two higher for the purpose of extracts per day. And their FCB reduces the mental ability penalty while under a mutagen.
Of the races you mention, it depends on your campaign. I played a tiefling alchemist up to 16th level in PFS and never had a problem running out of bombs after 3rd level or so. The bonus damage is great because it adds to your splash and sticky bomb damage as well. Unless you are doing a continuous dungeon crawl you probably don’t need the gnome FCB. Extra discoveries are awesome in any campaign, but think about levels. If you expect the campaign to end around level 10 or 11 that’s only one discovery.
If you just want to be small for aesthetic reasons, you could also just as your DM if it's okay if your character is under the effects of a permanencied reduce person spell.
I did that with my Strix. Cost about ~3000 gold and now I'm a tiny flying bombing machine.