Well, there are going to be some repeats here, but to have them all in the same place (although I might forget a few…)
I put the ones that are moderately to very unusual, but do exist, in green 
Birds
Boys
Ari / ”rni / Örn - Eagle
Gaukur – Cuckoo
Haukur - Hawk
Hrafn - Raven
Már - Gull
Orri - Black Grouse
Smyrill - Merlin
Svanur - Swan
Tjaldur - Oystercatcher
Valur - Falcon
Þröstur - Thrush
Girls
Arna / Erna - Eagle
Dúfa - Dove
Erla - Wagtail
Kría - Arctic Tern
Hrefna - Raven
Lóa - Golden Plover
Svala - Swallow
Svana - Swan
Ugla - Owl
There are various compound names for both sexes that start with Arn-, Hrafn- or Svan- also.
Animals
Boys
Björn / Bjarni / Bessi / Bersi- Bear
Hjörtur - Hart
Húnn - Bear-cub
Högni - Tom-cat
Kópur - Seal-pup
Mörður - Marten
Úlfur - Wolf
Girls
Birna - Bear
Ylfa - Wolf
There are also tons of names for boys and girls that use the elements Björn-, -björn, Bjarn-, Bjarnar-, Úlf-, -úlfur, -ólfur, -jólfur etc. Bears and wolves are very popular, especially bears.
Plants & Flowers
Boys
Burkni - Fern / Bracken
Börkur - Bark
Fífill - Dandelion
Liljar - Lily
Rósi - Rose
Smári - Clover
Girls
Dalía - Dahlia
Fífa - Cotton grass
Fjóla - Violet
Hvönn - Angelica
”ris - Iris
Lilja - Lily
Rós / Rósa - Rose
Sóley - Buttercup
There are loads of girls names that end with -rós or -lilja, also.
Other
Boys
Askur - Ash
Breki - Rock below the surface of the sea
Bjartur - Bright
Eldur - Fire
Fannar - Snow-Warrior
Frosti - Frost
Logi - Flame
Máni - Moon
Myrkvi - Darkness
Rökkvi - Dusk
Snær - Snow
Steinn - Stone
Sumarliði - Summer-Farer / Summer-Warrior
Sær - Sea
Tindur - Mountain peak
Veturliði - Winter-Farer / Winter-Warrior
Girls
Bára - Wave
Drífa - Snowfall
Dröfn - Wave
Dögg - Dew
Eyja - Island
Fönn - Snow
Lind - Spring
Njóla - Night
Nótt - Night
Mist - Mist
Mjöll - Snow
Ösp - Aspen
Prefixes on compound names ”s- (Ice), Snæ- (Snow), Fann- (Snow), Bjart- (Bright), Sól- (Sun), Stein- (Stone), Eld- (Fire), Ey- (Island). Suffixes on compound names -laug (Pool), -lind (Spring), -steinn (stone), -ey (island).
You can search about on the whole list here. All you have to know really is that stúlkur = girls and drengir = boys. If you see anything you like, I can look it up for you.