As GalacticCowboy's answer suggests, the phrase seems to be related to commissioning elders. Given that Paul was a student of Gamaliel (Acts 22) and the author of Hebrews seems deeply knowledgeable about Jewish sacrificial rites and uses arguments similar to Paul's in Galatians, it seems possible that both are referencing the rabbinic practice of semikhah to pass on authority. Wikipedia says there are three types of semikhah of which the first seem most relevant:
The recipient of this semikhah demonstrated sufficient education and proper judgment to be able to render halakhic judgments on matters of religious law as it pertains to daily life such as kashrut, nidda, and permissible or forbidden activities on Shabbos or Yom Tov.
(Don't ask me what all the Hebrew words are—I'd need to look them up.) Halakhic judgments include not only interpreting Jewish law, but applying it to day-to-day living. It's a big deal and a massive amount of authority. Paul's relationship to Timothy seems very much like the relationship between a senior rabbi and his disciple. According to Wikipedia, the passing of semikhah represented an unbroken chain of rabbis and elders all the way back to Moses.
The verb translated "laying on" (epithesis <1936>) is used another time in Acts 8:14-19 (NET):
Now when the apostles in Jerusalem heard that Samaria had accepted the word of God, they sent Peter and John to them. These two went down and prayed for them so that they would receive the Holy Spirit. (For the Spirit had not yet come upon any of them, but they had only been baptized in the name of the Lord Jesus.) Then Peter and John placed their hands on the Samaritans, and they received the Holy Spirit.
Now Simon, when he saw that the Spirit was given through the laying on of the apostles’ hands, offered them money, saying, “Give me this power too, so that everyone I place my hands on may receive the Holy Spirit.”
In this case, the practice somehow relates to baptism and receiving the Holy Spirit. While there are plenty of interpretations of this passage, placing hands was not reserved for elders, but offered to all believers. It seems to have closer ties to John the Baptist and the Essenes, though that's just my speculation. Even so, placing hands on the Samaritans represented a transmission of authority. The word translated "power" (exousia <1849>) has a connotation of authority.