About this phrase

Am is a masculine noun meaning time

Éigin is an adjective meaning some (in the sense of being unknown or unspecified — 'something''ll turn up, 'I'll get round to it some day' and so on).

Eile is an adjective meaning other.

Thus, am éigin eile: some other time.

Notes

In naturalistic speech éigin is pronounced as follows:

• Connacht: /əˈk′i:n′t′/.
• Munster: /ˈe:g′in′t′/.
• Ulster: /ənˈt′art/.

Note that all forms feature a terminal t.