Something that is worn out, cheap or insignificant. To give someone the rinky-dink means to cheat them.
The derivation of this expression isn't known, but we do know that it originated in the USA around the turn of the 20th century. Both of the above meanings, i.e. the adjectival meaning 'cheap and tacky' and the verb 'to cheat' originated then.
The earliest citation refer to the 'cheating' meaning; for example, this piece from The Fresno Morning Republican, June 1899:
But I - I ain't got nothln' for to show him what I think,
But you can bet I'll not set by and get the rinky-dink!
The 'cheap and tacky' meaning seems to have derived from that. The earliest citation I can find for that usage is in the Pennsylvania newspaper The McKean Democrat, February 1912:
"Some people will put a quarter column advertisement in some little rinky-dink sheet for $9 a year."