Here's the official policy on name availability:
Accounts that go unused for too long are susceptible to Summoner name decay. If an account hasn't played any games for a certain amount of time, their Summoner name will be flagged as available. The Summoner name remains on the account until another player claims it, at which point the Summoner name will be removed. If an inactive account loses its Summoner name this way, there will be a prompt to select a new Summoner name the next time that account logs into the client.
- Each Summoner name will have 6 months of inactivity protection upon creation.
- Each Summoner name will earn an additional month of inactivity protection for each Summoner level above 6, up to 30 months.
- Inactivity is defined to be a period in which no games of any kind have been played on the account.