The Collaborative International Dictionary of English v.0.48:

Balmy \Balm"y\, a.
   1. Having the qualities of balm; odoriferous; aromatic;
      assuaging; soothing; refreshing; mild; as, balmy weather.
      "The balmy breeze." --Tickell.
      [1913 Webster]

            Tired nature's sweet restorer, balmy sleep! --Young.
      [1913 Webster]

   2. Producing balm. "The balmy tree." --Pope.
      [1913 Webster]

   3. Highly eccentric or crazy. [Informal]

   Syn: batty.
        [PJC]

   Syn: Fragrant; sweet-scented; odorous; spicy; refreshing;
        soothing.
        [1913 Webster]

WordNet (r) 3.0 (2006):

balmy
    adj 1: informal or slang terms for mentally irregular; "it used
           to drive my husband balmy" [syn: balmy, barmy,
           bats, batty, bonkers, buggy, cracked,
           crackers, daft, dotty, fruity, haywire,
           kooky, kookie, loco, loony, loopy, nuts,
           nutty, round the bend, around the bend, wacky,
           whacky]
    2: mild and pleasant; "balmy days and nights"; "the climate was
       mild and conducive to life or growth"; "a soft breeze" [syn:
       balmy, mild, soft]

Moby Thesaurus II by Grady Ward, 1.0:

132 Moby Thesaurus words for "balmy":
   Saturnian, absurd, agreeable, allaying, alleviating, alleviative,
   ambrosial, analgesic, anesthetic, anodyne, aromatic, assuaging,
   assuasive, balsamic, bananas, barmy, bats, batty, beany, benumbing,
   bland, blooming, blossoming, bonkers, booming, bright, buggy,
   bughouse, bugs, cathartic, cleansing, clear, crackers, crazy,
   cuckoo, daffy, deadening, delightful, demulcent, dippy, dotty,
   dulling, easing, emollient, essenced, exuberant, faint, fair, fat,
   flaky, flipped, flourishing, flowering, flowery, fragrant,
   freaked-out, fruitcakey, fruiting, fruity, gaga, going strong,
   goofy, gratifying, halcyon, harebrained, haywire, in full swing,
   in good case, incense-breathing, insane, just plain nuts, kooky,
   lenient, lenitive, lightening, loony, loopy, mild, mitigating,
   mitigative, musky, numbing, nuts, nutty, odorate, odoriferous,
   odorous, off the hinges, off the track, off the wall, pain-killing,
   palliative, palmy, perfumed, perfumy, piping, pleasant, pleasing,
   potty, preposterous, prospering, purgative, redolent, refreshing,
   rejuvenating, relieving, remedial, restorative, rosy,
   round the bend, savory, scented, screwball, screwballs, screwy,
   silly, slaphappy, sleek, smooth, soft, softening, soothing, spicy,
   subduing, sunny, sweet, sweet-scented, sweet-smelling, thriving,
   thuriferous, vigorous, wacky