The verb απολυο, apoluo, means, according to Liddell & Scott (my one thousand page American 1864 edition) :
'to loose one thing from another' hence 'to set free or release from'.
Liddell & Scott then goes further and describes the main uses of the word to be :
'to acquit of a charge' or to 'release a prisoner'.
Thayer  has :
'set free, liberate' and 'let go, dismiss'
Απολυο is much stronger than the plain root λυο. This can be seen in its usage throughout scripture, for example when a synagogue, Acts 13:4, releases its congregation it is luo but when Jesus, Matthew 14:22, dismisses a congregation it is apoluo. (He has more authority than the Scribes and Pharisees, Matthew 7:29, thus attending one of his sermons is more binding and requires a stronger 'release' word.)
The words as far as I have studied them (in connection with redemption - lutrosis and apolutrosis) are very broad in concept, 'loosing' shoes, 'breaking' the sabbath, 'dismissing' congregations, and I eventually arrived, myself, at the word 'dispossess', for both luo and apoluo, which is actually what redemption is about.
But whatever word one arrives at for luo, apoluo is much, much stronger.
EGNT has 'release' in this place as does Young's Literal. KJV has 'set at liberty'. So they are being non-committal.
So why does the writer to Hebrews not say, precisely, what he means ?
There are four possibilities that I can think of :-
Timothy was under some kind of obligation (or discipline ?) within the church which was at an end.
Timothy was on a specific project or undertaking of some kind within the church which was now completed.
Timothy had some personal obligation which was now finished.
Timothy had been imprisoned or restricted in some way and was released.
Whatever it was, it was a private matter, personal to Timothy and to those around him and the writer wishes to allude to it, but prefers not to refer to it in detail.