LSJ has no entry for this headword, e)rai=a. (For 'branch' cf. epsilon 1384, kappa 1699. For 'refuse' cf. sigma 698.)
Assuming the headword is correctly transmitted, then the meaning given is a 'branch' or 'shoot' of a tree. If we also assume that our headword is cognate with the word for 'wool' [e)/rion], then any branch decorated with pieces of wool comes to mind. This interpretation has the support of ps.-Zonaras 865: writing later than the Suda, he adopted the Suda entry but added the words h)\ kai\ malli/on because he wished to make the connection with wool clear to his readers. See also ps.-Zonaras 869 (e)/rion. to\ malli/on) and Etymologicum Gudianum s.v. i(ereu/s : e)/rion ga\r le/getai to\ malli/on. See also epsilon 2929.
Such 'woolly' branches include the suppliant's branch and the eiresione or harvest-wreath. For the suppliant's branch see Aeschylus, Eumenides 43, Suppliant Women 22. [web addresses 1 & 2] and for the eiresione, epsiloniota 184. According to Crates the Athenian, the eiresione was a development from the suppliant's branch; see epsiloniota 184. Due to the prevalence of the olive in Attica the olive branch was principally used for these woolly branches. Compare Timaeus' Platonic Lexicon, who glosses the word qallo/s as 'any branch. But principally the branch of the olive' [pa=n to\ qallo/n. kuri/ws de\ o( th=s e)lai/as kla/dos]. On this hypothesis, the wool which distinguished the olive branch as the suppliant's branch or the eiresione led at some indeterminable time to the formation of our headword e)rai=a meaning a branch decorated with wool. We may compare in English the word 'sail', which can be used to refer to a 'ship', of which it forms the most prominent feature.
We learn further from epsiloniota 184 that, after the festival of the Pyanopsia, the eiresionai were placed at the doors of people's houses and left there to wither until they were replaced the following year. Aristophanes compared a wizened old women to an old eiresione that a spark could set alight: see Wealth [Plutus] 1054 (web address 3). Consequently, the Suda would seem to be saying that the word e)rai=a also came to refer to any withered branch of an olive tree, disregarding the association with wool.