What about Aisling or Aislinn -- that'd get your "lin" in. It's means something like "dream", I guess..daydreaming is "aislingeacht".

Note: the "i" next to the "s" will make a "sh" sound (a voiceless, alveolar fricative -- to be a know-it-all). Therefore, the name's pronounced more like Ashling or Ashlynn depending.