Travelling lane gets old on one’s own (5)
I believe the answer is:
alone
'on one's own' is the definition.
(I've seen this in another clue)
'travelling lane gets old' is the wordplay.
'travelling' indicates anagramming the letters.
'gets' means one lot of letters goes inside another (gets can mean captures or absorbs).
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'lane' with letters rearranged gives 'alne'.
'alne' going around 'o' is 'ALONE'.
(Other definitions for alone that I've seen before include "On one's own" , "unique" , "Solo" , "Without others being involved" , "Without others present" .)