What does Egan mean?
Anglicized form of the surname Mac Aodhagain, 'son of Aodhagan', where Aodhagan is a double diminutive of Aodh ('fire'). The fire sense is inherited from that root rather than the name itself.
What does Tynan mean?
Anglicized from Irish O Teimhneain, from a diminutive of Old Irish 'teimhean' meaning 'dark, dusky, gloom'. (A minority source instead links it to O Tadhgain, 'poet', but the 'dark' derivation is the one usually given.)